Vertex Spans of Multilayered Cycle and Path Graphs

Abstract

In this paper, we observe a special class of graphs known as multilayered graphs and their subclasses, namely multilayered cycles and multilayered paths. These graphs model layouts of shopping malls, city street grids, and even resemble the topology of certain famous board games. We analyze the values of all vertex spans (strong, direct, and Cartesian span) for these subclasses of graphs. Surprisingly, our results for multilayered cycles reveal that, regardless of the chosen movement rules, the span values depend solely on the length of the individual cycles, rather than the number of layers. This finding carries significant implications for the application of graph spans in maintaining safety distances.

1. Introduction and Motivation

In response to the widespread adoption of social distancing measures during recent pandemic years, Banič and Taranenko used the concept of “span” from topology [1] and developed a graph theoretical measure known as the “graph span” [2]. At its core, it is the maximal safety distance that two players can keep while moving through the vertices of some graph. Basic definitions introduce six types of graph spans, depending on weather the players have to visit all the vertices or all the edges of a graph (vertex and edge span) and what movement rules they follow (strong, direct, and Cartesian span). In our previous research [3], we explored the relationships between different types of vertex spans and determined span values for specific graph classes. Additionally, in [4], we conducted an analysis of edge spans and the minimum lengths of walks required to achieve these spans. In this paper, we once again observe vertex spans, now for two special classes of graphs—multilayered cycles, denoted by $M{C}_n^k$, and multilayered paths, denoted by $M{P}_n^k$. We can imagine a multilayered cycle as k isomorphic cycles $C_n$, stacked on top of one another, with the corresponding vertices joined by “vertical” edges, forming a cylinder shape while multilayered paths form a grid graph with n vertices “horizontally” and k vertices “vertically”. These graphs are isomorphic to Cartesian products of path $P_k$ and cycle $C_n$, or two paths, $P_n$ and $P_k$, respectively [5]; however, we use the terms multilayered cycle and multilayered path to better correspond with our motivation. In graph theory research, there are many directions in which the notion of distance is fundamental, from the classical observations to some newer ones [6,7,8]. In fact, in the recent years, many research papers focused on observing the game theory approach in the problems of distances in graphs [9,10,11,12]. There is also some research on these classes of graphs we observe; for instance, weak homomorphisms on a stacked prism graph, which is another name for a multilayered cycle, or shortest paths on grid graphs [13,14]. Our motivation to observe vertex spans for these graph class stems from several distinct sources: first the need to determine safe occupancy limits in shopping malls (which are often designed in the multilayered cycle configuration), in response to social distancing measures, and second from our interest piqued by graph-based games, such as “Cops and Robbers” games [15]. There are different versions of “Cops and Robbers” games, but the main idea is to have at least one “robber” and at least one “cop” moving through graph vertices, while the cop is trying to “catch” the robber and the robber is trying to keep his distance from the cop. There are many versions of the game where multilayered cycles are a common playground. Observed from above, it resembles a spider’s web, for instance, the game “The Spider and The Flies”, developed in 1898 (also “Web Chase” and “The Spider’s Web”). It is also worth noting that a version of a well-known game “Nine man’s morris” called “Morabaraba” (or “Twelve men’s morris”), which is played in South Africa as a sport, is played on a board that is exactly a multilayered cycle (Figure 1).

Grids, on the other hand, find applications in various scenarios such as representations of city streets. Consequently, the concept of safety distance holds implications, especially if there’s a necessity to regulate the presence of a fixed number of individuals within a bounded area of the city. Additionally, grid-based structures are ubiquitous in various games where players may either aim to maintain distance from one another or actively pursue each other, as seen in games like “Cops and Robbers”.

In this paper, we provide a mathematical foundation for understanding span values in multilayered cycles and multilayered paths. Future research could delve deeper into the mathematical analysis of the games inspired by these structures as well as the generalization of the concept of span to accommodate scenarios involving three or more players navigating the graph. Our results on multilayered cycles demonstrate that, regardless of the chosen movement rules, span values for multilayered cycles are solely dependent on the cycle’s length, rather than the cylinder’s height. This finding is particularly intriguing and has significant practical implications. For multilayered paths $M{P}_n^k$, we found that all the vertex span values depend only on the lesser of the two values n and k. In Section 2, basic definitions and preliminaries for our research are given, and in Section 3 and Section 4, we present our main findings for multilayered cycles and multilayered paths. Section 5 summarizes our results, while in Section 6, we offer the open problems, ideas and directions for possible future research.

2. Preliminaries and Definitions

The term graph refers to a simple connected graph in the rest of the paper. We use standard graph theory notation [16].

Definition 1.

We let G be a graph with n vertices and $k\in \mathbb{N},k\ge 2$. Multilayered graph $M{G}^k$ is a graph with $nk$ vertices denoted by $(i,j)$ where $i\in V\left(G\right)$ and $j\in {\mathbb{N}}_k$ and in which $\left((a,b)(c,d)\right)\in E\left(M{G}^k\right)$ if one of the following holds:

• $a=c$ and $|d-b|=1$; or
• $b=d$ and $ac\in E\left(G\right)$.

The example of a multilayered graph is shown in Figure 2.

Readers more familiar with graph products will notice that multilayered graph $M{G}^k$ is actually a Cartesian product of graph G and path graph $P_k$, i.e., $M{G}^k=G\square P_k$.

In this paper, we observe special classes of multilayered graphs which are multilayered cycles and multilayered paths, i.e., multilayered graphs $M{G}^k$ where graph G is either cycle $C_n$ or path $P_n$.

Definition 2.

We let $n,k\in \mathbb{N},n\ge 3,k\ge 2$. Multilayered cycle $M{C}_n^k$ is a graph with $nk$ vertices denoted by $(i,j)$ where $i\in {\mathbb{Z}}_n$ and $j\in {\mathbb{N}}_k$, and in which $\left((a,b)(c,d)\right)\in E\left(M{C}_n^k\right)$ if one of the following holds:

• $a=c$ and $|d-b|=1$; or
• $b=d$ and $|a-c|\in \{1,n-1\}$.

Definition 3.

We let $n,k\in \mathbb{N},n\ge 2,k\ge 2$. Multilayered path $M{P}_n^k$ is a graph with $nk$ vertices denoted by $(i,j)$ where $i\in {\mathbb{N}}_n$ and $j\in {\mathbb{N}}_k$, and in which $\left((a,b)(c,d)\right)\in E\left(M{P}_n^k\right)$ if one of the following holds:

• $a=c$ and $|d-b|=1$; or
• $b=d$ and $|a-c|=1$.

We refer to vertex $(i,j)$ as vertex i in layer j. Also, we denote the layer of vertex $(i,j)$ by $p_2(i,j):=j$, as a layer is but a projection of a vertex to the second coordinate. Note that multilayered cycles are also known as stacked prism graphs or “web” graphs and multilayered paths as grid graphs or lattice graphs. Also, cube graph $Q_3$ is isomorphic to multilayered cycle $M{C}_4^2$.

The examples of a multilayered cycle and a multilayered path are given in Figure 3.

To describe the movement of two players in a graph, we define three types of functions that map ${\mathbb{N}}_l$, for some $l\in \mathbb{N}$, to the set of graph vertices [3]. Such a function represents the movement of a player through graph vertices in l steps. These functions, and consequently vertex spans, are defined corresponding to three different movement rules that two players can apply in a graph, so let us repeat those rules.

• Traditional movement rules: Both players move independently of one another; one can stand still while the other one moves, or they can both move at the same time;
• Active movement rules: Both players move to an adjacent vertex in each step;
• Lazy movement rules: In each step, exactly one of the players moves to an adjacent vertex while the other stands still.

The definitions of l-tracks, lazy l-tracks and opposite lazy l-tracks are given in [3], but since they are of great importance for this paper, we repeat them here.

Definition 4.

We let $G=(V,E)$ be a graph and $l\in \mathbb{N}$. We say that surjective function $f_l:{\mathbb{N}}_l⟶V\left(G\right)$ is an l-track on G if $f\left(i\right)f(i+1)\in E\left(G\right)$ holds for each $i\in {\mathbb{N}}_{l-1}$.

Definition 5.

We let $G=(V,E)$ be a graph and $l\in \mathbb{N}$. We say that surjective function $f_l:{\mathbb{N}}_l⟶V\left(G\right)$ is a lazy l-track on G if $f\left(i\right)f(i+1)\in E\left(G\right)$ or $f\left(i\right)=f(i+1)$ holds for each $i\in {\mathbb{N}}_{l-1}$.

Definition 6.

We let G be a graph, $f,g:{\mathbb{N}}_l⟶V\left(G\right)$ lazy l-tracks on G. We say that f and g are opposite lazy l-tracks on G if

$$f\left(i\right)f(i+1)\in E\left(G\right)\iff g\left(i\right)=g(i+1)$$

(1)

for all $i\in {\mathbb{N}}_{l-1}$.

For reasons of simplifying our proofs, we use the following terminology for lazy l-tracks in multilayered cycles and paths regarding the images of consequent steps. We let G be a multilayered cycle or a multilayered path, $l\in \mathbb{N}$ and $f:{\mathbb{N}}_l⟶V\left(G\right)$ a lazy l-track. For both multilayered cycles and paths, we employ the following terminology:

• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x,y)$, we say that f stands still in step i;
• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x,y+1)$, we say that f moves up in step i;
• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x,y-1)$, we say that f moves down in step i.

Solely for multilayered cycles, we employ the following terminology:

• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x{+}_{n}1,y)$, we say that f moves counter-clockwise in step i;
• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x{-}_{n}1,y)$, we say that f moves clockwise in step i.

Solely for multilayered paths, we employ the following terminology:

• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x+1,y)$, we say that f moves right in step i;
• If $f\left(i\right)=(x,y)$ and $f(i+1)=(x-1,y)$ we say that f moves left in step i.

Note that those are the only options for any lazy l-track on $M{C}_n^k$ and $M{P}_n^k$. If for some $i\in {\mathbb{N}}_l$ and a lazy l-track f on $M{C}_n^k$ or $M{P}_n^k$, we have $f\left(i\right)=(a,b)$, we say that f is in layer b.

In order to formally define vertex spans, i.e., the maximal safety distance, we first provide the definition of the distance between two lazy l-tracks.

Definition 7.

We let G be a graph, $l\in \mathbb{N}$ and $f,g$ two lazy l-tracks on G. We define

$$m_G(f,g)=min\{d(f\left(i\right),g\left(i\right):i\in {\mathbb{N}}_l\}$$

to be the distance between f and g.

Analogously to an l-sweepable graph in [4], we define an l-trackable graph.

Definition 8.

We let G be a graph and $l\in \mathbb{N}$. If at least one lazy l-track exists on G, we say that G is an l-trackable graph.

Lastly, we offer definitions for different vertex spans, first described in [2] and then redefined in [3].

We let G be an l-trackable graph. We define

$$M_l^{⊠}:=max\{m_G(f,g):f\mathrm{and}g\mathrm{are}\mathrm{lazy}l-\mathrm{tracks}\mathrm{on}G\}.$$

(2)

$$M_l^{\times }:=max\{m_G(f,g):f\mathrm{and}g\mathrm{are}l-\mathrm{tracks}\mathrm{on}G\}.$$

(3)

We let G be a graph and $l\in \mathbb{N}$ such that at least one pair of opposite lazy l-tracks exists on G. We define

$$M_l^{\square }:=max\{m_G(f,g):f\mathrm{and}g\mathrm{are}\mathrm{opposite}\mathrm{lazy}l-\mathrm{tracks}\mathrm{on}G\}.$$

(4)

We let G be a graph and let $S\subseteq \mathbb{N}$ be the set of all integers l for which G is an l-trackable graph. We define the strong vertex span as the number

$${\sigma }_V^{⊠}\left(G\right):=max\{M_l^{⊠}:l\in S\}.$$

(5)

This number is the maximal safety distance that can be kept while two players visit all the vertices of a graph while following the traditional movement rules.

We define the direct vertex span as the number

$${\sigma }_V^{\times }\left(G\right):=max\{M_l^{\times }:l\in S\}.$$

(6)

This number is the maximal safety distance that can be kept while two players visit all the vertices of a graph with respect to the active movement rules.

We let G be a graph and let $C\subseteq \mathbb{N}$ be the set of all integers l for which opposite lazy l-tracks exist on G. We define the Cartesian vertex span as the number

$${\sigma }_V^{\square }\left(G\right):=max\{M_l^{\square }:l\in C\}.$$

(7)

This number is the maximal safety distance that can be kept while two players visit all the vertices of a graph with respect to the lazy movement rules.

3. Results on Multilayered Cycles

We now proceed with the results for vertex span values for multilayered cycles.

Lemma 1.

We let graph $G=M{C}_n^k$ for some $n,k\in \mathbb{N}$. Also, we let $f,g$ be two opposite lazy l-tracks on G, $l\in \mathbb{N}$. Then, there exists $i\in {\mathbb{N}}_l$ such that $p_2\left(f\left(i\right)\right)=p_2\left(g\left(i\right)\right)$, i.e., $f\left(i\right)$ and $g\left(i\right)$ are in the same layer.

Proof. 

We let $f,g$ be any two opposite lazy l-tracks on G. If $f\left(1\right)$ and $g\left(1\right)$ are in the same layer, then the claim holds. Otherwise, let us assume that $f\left(1\right)$ is in layer x and $g\left(1\right)$ is in layer $y\ne x$. Without any loss of generality, we can assume that $x<y$, so $x-y<0$. Since f is surjective, there exists $j\in {\mathbb{N}}_l$ such that $f\left(j\right)$ is in the layer k. For such j, it holds that $p_2\left(f\left(j\right)\right)-p_2\left(g\left(j\right)\right)=k-p_2\left(g\left(j\right)\right)\ge 0$. Since f and g are opposite, if $p_2\left(f\left(b\right)\right)-p_2\left(g\left(b\right)\right)=a$, for some $a,b\in \mathbb{N}$, then $p_2\left(f(b+1)\right)-p_2\left(g(b+1)\right)\in \{a-1,a,a+1\}$. Now, given the fact that $p_2\left(f\left(1\right)\right)-p_2\left(g\left(1\right)\right)<0$, $p_2\left(f\left(j\right)\right)-p_2\left(g\left(j\right)\right)\ge 0$ and that the difference between layers changes by at most one for consequent steps, we know that there must exist some $i\in \{2,...,j\}$ such that $p_2\left(f\left(i\right)\right)-p_2\left(g\left(i\right)\right)=0$ and therefore $p_2\left(f\left(i\right)\right)=p_2\left(g\left(i\right)\right)$. □

An example of one such movement is presented in Figure 4.

Lemma 2.

${\sigma }_V^{\square }\left(M{C}_n^k\right)\le ⌊\frac{n}{2}⌋$.

Proof. 

Since by Lemma 1, for any two opposite lazy l-tracks, f and g, there exists an $i\in {\mathbb{N}}_l$ such that $p_2\left(f\left(i\right)\right)=p_2\left(g\left(i\right)\right)$, then for such i, $d(f\left(i\right),g\left(i\right))\le diam\left(C_n\right)=⌊\frac{n}{2}⌋$, hence ${\sigma }_V^{\square }\left(M{C}_n^k\right)\le ⌊\frac{n}{2}⌋$. □

Theorem 1.

${\sigma }_V^{\square }\left(M{C}_n^k\right)=⌊\frac{n}{2}⌋$.

Proof. 

First, we construct two lazy l-tracks, f and g, where $l=2nk-1$, that are always at a distance of at least $⌊\frac{n}{2}⌋$. We start by defining $f\left(1\right)=(0,1)$ and $g\left(1\right)=(⌊\frac{n}{2}⌋,1)$, and then proceed in the following way: on odd steps, f moves and g stands still, and on even steps, f stands still and g moves. We define all movements in four stages.

• For the first $2k-2$ steps, on odd ones, f moves up, and on even ones, g moves up. So $f(2k-1)=(0,k)$ and $g(2k-1)=(⌊\frac{n}{2}⌋,k)$.
• On the next two steps, first, f moves counter-clockwise, and then g moves counter-clockwise. So, $f(2k+1)=(1,k)$ and $g(2k+1)=(1+⌊\frac{n}{2}⌋,k)$.
• For the next $2k-2$ steps, on odd ones, f moves down, and on even ones, g moves down. So $f(4k-1)=(1,1)$ and $g(4k-1)=(1+⌊\frac{n}{2}⌋,1)$.
• On the next two steps, first, f moves counter-clockwise, and then g moves counter-clockwise.

Now, we repeat these four stages of movement until all vertices are visited. This kind of movement is presented in Figure 5. It is easily seen that, defined this way, f and g are always at a distance of at least $⌊\frac{n}{2}⌋$. So, ${\sigma }_V^{\square }\left(M{C}_n^k\right)\ge ⌊\frac{n}{2}⌋$. Combined with Lemma 2, we have ${\sigma }_V^{\square }\left(M{C}_n^k\right)=⌊\frac{n}{2}⌋$.

□

Lemma 3.

We let graph $G=M{C}_n^k$ for some $n,k\in \mathbb{N}$. Also, we let $f,g$ be two lazy l-tracks on G, $l\in \mathbb{N}$. Then, there exists $i\in {\mathbb{N}}_l$ such that $|p_2\left(f\left(i\right)\right)-p_2\left(g\left(i\right)\right)|\le 1$, i.e., $f\left(i\right)$ and $g\left(i\right)$ are either in the same or in adjacent layers.

Proof. 

Much like in the proof of Lemma 1, we can easily see that if, for some $b\in {\mathbb{N}}_l$, $|p_2\left(f\left(b\right)\right)-p_2\left(g\left(b\right)\right)|=a$, then $|p_2\left(f(b+1)\right)-p_2\left(g(b+1)\right)|\in \{a-2,a-1,a,a+1,a+2\}$. In other words, in each step, the difference between layers of $f\left(b\right)$ and $g\left(b\right)$ can change by at most 2. The same line of reasoning as in the proof of Lemma 1 leads us to conclusion that, for some $i\in {\mathbb{N}}_l$, $f\left(i\right)$ and $g\left(i\right)$ are either in the same or in the adjacent layers. □

An example of one such movement is presented in Figure 6.

Lemma 4.

${\sigma }_V^{⊠}\left(M{C}_n^k\right),{\sigma }_V^{\times }\left(M{C}_n^k\right)\le ⌊\frac{n}{2}⌋+1$.

Proof. 

By Lemma 3, for any two lazy l-tracks f and g, there exists an $i\in {\mathbb{N}}_l$ such that $f\left(i\right)$ and $g\left(i\right)$ are in the same or neighbouring layers. If $f\left(i\right)$ and $g\left(i\right)$ are in the same layer, then $d(f\left(i\right),g\left(i\right))\le diam\left(C_n\right)=⌊\frac{n}{2}⌋$, hence both ${\sigma }_V^{⊠}\left(M{C}_n^k\right)$ and ${\sigma }_V^{\times }\left(M{C}_n^k\right)$ are less than $⌊\frac{n}{2}⌋+1$. If, on the other hand, $f\left(i\right)$ and $g\left(i\right)$ are in the neighbouring layers, then we can assume, without any loss of generality, that $f\left(i\right)$ is in one layer above $g\left(i\right)$, so $f\left(i\right)=(a,b)$ and $g\left(i\right)=(c,b-1)$ for some $a,b,c\in \mathbb{N}$. Now, $d(f\left(i\right),g\left(i\right))=d((a,b),(c,b))+1\le ⌊\frac{n}{2}⌋+1$. □

Theorem 2.

${\sigma }_V^{⊠}\left(M{C}_n^k\right)={\sigma }_V^{\times }\left(M{C}_n^k\right)=⌊\frac{n}{2}⌋+1$.

Proof. 

We construct two l-tracks f and g that start on the distance $⌊\frac{n}{2}⌋+1$ and keep that distance at all times. First, let us describe the l-track, f. It starts in vertex $(0,1)$, moves through the whole layer clockwise, then moves up one layer and proceeds through the whole layer again in the same way. It continues to do so until it reaches the topmost layer and moves through it as well. Lastly, it moves down one layer and once again moves through it clockwise. For graph $M{C}_6^4$, this movement is presented in Figure 7.

It is easily seen that this way, f visits all the vertices. Now, we describe the movement of g depending on the movement of l-track f. l-track g starts its movement in vertex $(⌊\frac{n}{2}⌋,2)$, so $d(f\left(1\right),g\left(1\right))=⌊\frac{n}{2}⌋+1$. Whenever l-track f moves clockwise, g also moves clockwise, thus maintaining the same distance as well as visiting its whole layer while f is visiting its own. The first time that f moves up, g moves down, and afterwards, whenever f changes layers, g moves up. This way, g visits all the layers, and at each one it moves through all of its vertices while maintaining the same distance at all times. For graph $M{C}_6^4$, the movement of g is presented in Figure 8.

It is easily seen that, defined this way, that f and g are always at a distance of at least $⌊\frac{n}{2}⌋+1$. So, both ${\sigma }_V^{⊠}\left(M{C}_n^k\right)$ and ${\sigma }_V^{\times }\left(M{C}_n^k\right)$ are greater than or equal to $⌊\frac{n}{2}⌋+1$. Combined with Lemma 4, we have ${\sigma }_V^{⊠}\left(M{C}_n^k\right)={\sigma }_V^{\times }\left(M{C}_n^k\right)=⌊\frac{n}{2}⌋+1$. □

4. Results on Multilayered Paths

We now proceed with the results for vertex span values for multilayered paths.

Lemma 5.

We let graph $G=M{P}_n^k$,= for some $n,k\in \mathbb{N}$. Also, we let $f,g$ be two opposite lazy l-tracks on G, $l\in \mathbb{N}$. Then, there exists $i\in {\mathbb{N}}_l$ such that $p_2\left(f\left(i\right)\right)=p_2\left(g\left(i\right)\right)$, i.e., $f\left(i\right)$ and $g\left(i\right)$ are in the same layer.

Proof. 

Proof of this Lemma is completely analogous to the one of Lemma 1. □

For further considerations, we observe that $M{P}_n^k$ is isomorphic to $M{P}_k^n$, since $M{P}_n^k=P_n\square P_k$ and the Cartesian product of graphs is commutative.

Lemma 6.

${\sigma }_V^{\square }\left(M{P}_n^k\right)\le min\{n,k\}-1$.

Proof. 

Without the loss of generality, we let $n\le k$. Since by Lemma 5, for any two opposite lazy l-tracks f and g, there exists some $i\in {\mathbb{N}}_l$ such that $p_2\left(f\left(i\right)\right)=p_2\left(g\left(i\right)\right)$, then for such i, $d(f\left(i\right),g\left(i\right))\le diam\left(P_n\right)=n-1$; hence, ${\sigma }_V^{\square }\left(M{P}_n^k\right)\le n-1$. We use the same reasoning for $k\le n$ and determine that ${\sigma }_V^{\square }\left(M{P}_n^k\right)$ has to be less than or equal to $min\{n,k\}-1$. □

Theorem 3.

${\sigma }_V^{\square }\left(M{P}_n^k\right)=min\{n,k\}-1$.

Proof. 

Let us assume that $n\le k$ and prove that, in that case, ${\sigma }_V^{\square }\left(M{P}_n^k\right)=n-1$.

First, we construct two opposite lazy l-tracks, f and g, that are always at a distance of at least $n-1$. We start by defining $f\left(1\right)=(1,1)$ and $g\left(1\right)=(n,k)$, and then proceed in the following way: First, g stands still while f visits all the vertices that are on the distance of at least $n-1$ from vertex $(n,k)$ (first two grids in Figure 9). Now, f moves back to $(1,1)$. Next, f stands still while g visits all vertices that are on the distance at least $n-1$ from vertex $(1,1)$ and then moves back to its starting position (the third grid in Figure 9). Then, f and g swap places by moving along the outer rim of a grid graph. They move one at a time and at all times keep their distance at at least $n-1$. Lastly, they visit the vertices that they previously did not in the same fashion they did in the first part of the movement. The swapping of positions and the last movement are presented in the last two grids in Figure 9. □

For the direct vertex span, we first provide the results for the square grid graph, i.e., for $M{P}_n^n$. Afterwards, we use those results to prove the claim for direct and strong vertex span values in the general case.

Lemma 7.

It holds that ${\sigma }_V^{\times }\left(M{P}_n^n\right)\le 2⌊\frac{n}{2}⌋$.

Proof. 

The claim follows from the fact that all vertex span values of any graph must necessarily be less than or equal to its radius, which was proven in Lemma 2.5 in [3]. It is easy to see that

$$rad\left(M{P}_n^k\right)=⌊\frac{n}{2}⌋+⌊\frac{k}{2}⌋,$$

so it follows that

$$rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋=\left\{\begin{array}{cc}n-1,\hfill & n\mathrm{odd};\\ n,\hfill & n\mathrm{even}.\end{array}\right.$$

Hence, ${\sigma }_V^{\times }\left(M{P}_n^n\right)\le 2⌊\frac{n}{2}⌋$. □

Theorem 4.

It holds that ${\sigma }_V^{\times }\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋$.

Proof. 

We construct two l-tracks, f and g, in $M{P}_n^n$ such that their distance is always equal to or greater than the radius of that graph. First, let us note the following. For vertices $(n,n),(a,b)\in V\left(M{P}_n^n\right)$, where $a,b\le ⌈\frac{n}{2}⌉$ it holds that

$$d((n,n),(a,b))\ge rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋.$$

Also, for vertices $(n-1,n),(c,d)\in V\left(M{P}_n^n\right)$, where $c,d\le ⌈\frac{n}{2}⌉$ and $(c,d)\ne \left(⌈\frac{n}{2}⌉,⌈\frac{n}{2}⌉\right)$ it holds that

$$d((n-1,n),(c,d))\ge rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋.$$

Analogously, for the same vertex $(c,d)$, it holds that

$$d((n,n-1),(c,d))\ge rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋.$$

This means that all the vertices in subgraph $M{P}_{\lceil \frac{n}{2}\rceil }^{⌈\frac{n}{2}⌉}$ of graph $M{P}_n^n$ (the bottom left square subgraph) are at the distance of at least $2⌊\frac{n}{2}⌋$ from vertex $(n,n)$ (the upper left corner) in graph $M{P}_n^n$. Analogously, for each of the remaining three vertex corners, $(1,1),(1,n)$ and $(n,1)$, there is a square subgraph of dimensions $⌈\frac{n}{2}⌉\times ⌈\frac{n}{2}⌉$ such that all its vertices are on the distance at least $2⌊\frac{n}{2}⌋$ from the corresponding corner vertex. Moreover, vertices $(n-1,n)$ and $(n,n-1)$ that are adjacent to the corner vertex are on the distance of at least $2⌊\frac{n}{2}⌋$ from all the vertices in subgraph $M{P}_{\lceil \frac{n}{2}\rceil }^{⌈\frac{n}{2}⌉}$, except for vertex $\left(⌈\frac{n}{2}⌉,⌈\frac{n}{2}⌉\right)$.

Let us describe l-tracks f and g.

We let $f\left(1\right)=(1,1)$ and $g\left(1\right)=(n,n)$. In the first stage, function g alternates between vertices $(n,n)$ and $(n-1,n)$ in such a way that $g\left(i\right)=(n,n)$ for even i and $g\left(i\right)=(n-1,n)$ for odd i. Meanwhile, f moves through all the vertices of subgraph $M{P}_{\lceil \frac{n}{2}\rceil }^{⌈\frac{n}{2}⌉}$. Note that f always moves from $(1,1)$ to $\left(⌈\frac{n}{2}⌉,⌈\frac{n}{2}⌉\right)$ in an even number of steps, so for any j for which $f\left(j\right)=\left(⌈\frac{n}{2}⌉,⌈\frac{n}{2}⌉\right)$, $g\left(j\right)=(n,n)$ holds, thus securing that $d(f\left(j\right),g\left(j\right))=rad\left(M{P}_n^n\right)$. This movement is shown in the first grid of Figure 10 and Figure 11.

Now, f and g move following the outer rim of the graph, g moves left via path $(n,n),(n-1,n),(n-2,n),...,(1,n)$, and f moves right through path $(1,1),(2,1),...,(n,1)$. Let us prove that while moving through this stage, their distance is at least $rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋$. Namely, for even n, f and g will never be in the same vertical path, i.e., there is no $j\in \mathbb{N}$ in this stage such that $f\left(j\right)=(x,1)$ and $g\left(j\right)=(x,n)$, so in each step i of this stage, $d(f\left(i\right),g\left(i\right))\ge diam\left(P_n\right)+1=n$. For odd n, there exists $j\in \mathbb{N}$ such that $f\left(j\right)=(x,1)$ and $g\left(j\right)=(x,n)$, so $d(f\left(j\right),g\left(j\right))=diam\left(P_n\right)=n-1$. Obviously, for all the other steps of this stage, the distance of f and g is greater than $diam{P}_n$. This stage of the movement of f and g is shown in the second grid in Figure 10 and Figure 11.

After the second stage, g is in the corner vertex $(1,n)$, and f is in the corner vertex $(n,1)$, and the movement proceeds analogously as in the first stage with g alternating between $(1,n)$ and any of its adjacent vertices. Continuing in this fashion two more times, f visits all the vertices, and by switching their roles and repeating all the steps, g also visits all the vertices. We show that for l-tracks f and g constructed in this way, $d(f,g)=rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋$ holds, so the claim is proven for the direct vertex span. □

Corollary 1.

It holds that ${\sigma }_V^{⊠}\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋$.

Proof. 

From [3], we know that for any graph G, it holds that ${\sigma }_V^{⊠}\left(G\right)\ge max\{{\sigma }_V^{\square }\left(G\right),{\sigma }_V^{\times }\left(G\right)\}$, and ${\sigma }_V^{⊠}\left(G\right),{\sigma }_V^{\square }\left(G\right),{\sigma }_V^{\times }\left(G\right)\le rad\left(G\right)$, so from ${\sigma }_V^{\times }\left(M{P}_n^n\right)=rad\left(M{P}_n^n\right)$, it follows that ${\sigma }_V^{⊠}\left(M{P}_n^n\right)=rad\left(M{P}_n^n\right)=2⌊\frac{n}{2}⌋$. □

The values of the direct and therefore strong vertex span of $M{P}_n^k$ when $n\ne k$ do not depend on the parity of n and k. Let us assume that $n<k$, since $M{P}_n^k$ is isomorphic to $M{P}_k^n$. We show that for for even n, the value for the strong span equals the value for the direct span and they equal n, which is the same as the value of the direct span for graph $M{P}_n^n$. For odd n, on the other hand, the value for the strong span again equals the value for the direct span and again they equal n; however, this differs from the direct span value of $M{P}_n^n$, which equals $n-1$.

First, we prove the upper limit for ${\sigma }_V^{\times }\left(M{P}_n^k\right)$, which is generally smaller than the graph radius.

Lemma 8.

We let graph $G=M{P}_n^k$, for some $n,k\in \mathbb{N}$ such that $n<k$. Also, we let $f,g$ be two lazy l-tracks on G, $l\in \mathbb{N}$. Then, there exists $i\in {\mathbb{N}}_l$ such that $|p_2\left(f\left(i\right)\right)-p_2\left(g\left(i\right)\right)|\le 1$, i.e., $f\left(i\right)$ and $g\left(i\right)$ are either in the same or in adjacent layers.

Proof. 

We let $G=M{P}_n^k$ for some $n,k\in \mathbb{N}$ such that $n<k$, and we let $f,g$ be two l-tracks on G, $l\in \mathbb{N}$. If $f\left(1\right)$ and $g\left(1\right)$ are in the same layer, then $|p_2\left(f\left(1\right)\right)-p_2\left(g\left(1\right)\right)|=0$. Let us assume that $f\left(1\right)$ and $g\left(1\right)$ are in different layers, and without the loss of generality we assume that $p_2\left(f\left(1\right)\right)<p_2\left(g\left(1\right)\right)$. Since f and g are surjective, they must visit all the layers, so there exists some $i\in {\mathbb{N}}_l$ such that $p_2\left(f\left(i\right)\right)\ge p_2\left(g\left(i\right)\right)$. We let j be the smallest number such that $p_2\left(f\left(j\right)\right)\ge p_2\left(g\left(j\right)\right)$. If $p_2\left(f\left(j\right)\right)=p_2\left(g\left(j\right)\right)$, then the claim is proven. If $p_2\left(f\left(j\right)\right)>p_2\left(g\left(j\right)\right)$, then $p_2\left(f(j-1)\right)<p_2\left(g(j-1)\right)$ must hold due to the minimality of number j. But this means that $|p_2\left(f\left(j\right)\right)-p_2\left(g\left(j\right)\right)|=|p_2\left(f(j-1)\right)-p_2\left(g(j-1)\right)|=1$, i.e., in steps $j-1$ and j, f and g are in the adjacent layers. □

Lemma 9.

${\sigma }_V^{⊠}\left(M{P}_n^k\right),{\sigma }_V^{\times }\left(M{P}_n^k\right)\le min\{n,k\}$.

Proof. 

Without the loss of generality, we let $n<k$ and let $f,g$ be lazy l-tracks on $M{P}_n^k$. By Lemma 8, for some $i\in {\mathbb{N}}_l$, f and g are in the same or in adjacent layers. If they are in the same layer for some $i\in {\mathbb{N}}_l$, then $d(f\left(i\right),g\left(i\right))\le diam{P}_n=n-1$, and if they are in adjacent layers, their distance is larger by 1, i.e., $d(f\left(i\right),g\left(i\right))\le diam{P}_n+1=n$. □

Theorem 5.

${\sigma }_V^{⊠}\left(M{P}_n^k\right),{\sigma }_V^{\times }\left(M{P}_n^k\right)=min\{n,k\}$.

Proof. 

We let $n<k$. In Lemma 9, we show that the direct and strong vertex span values cannot be larger than n, so it remains to describe l-tracks f and g for which the span values are achieved. We first prove that ${\sigma }_V^{\times }\left(M{P}_n^k\right)=min\{n,k\}=n$. We observe the parity of n in two separate cases.

• n is even. To construct l-tracks f and g such that $d(f,g)=n$, we make use of the l-tracks from the proof of Theorem 4. It is easy to see that graph $M{P}_n^n$ is a subgraph of $M{P}_n^k$. Let us denote by $f_1$ and $g_1$ the $l_1$-tracks in $M{P}_n^n$ such that $d(f_1,g_1)={\sigma }_V^{\times }\left(M{P}_n^n\right)=n$. To obtain functions f and g, we first expand functions $f_1$ and $g_1$ so that after they both visit all the vertices in subgraph $M{P}_n^n$ (the first grid in Figure 12), they return to the vertices they began in, $(1,1)$ and $(n,n)$.

  Figure 12. Example of movement for f and g in $M{P}_4^7$ for Case 1.

  Figure 12. Example of movement for f and g in $M{P}_4^7$ for Case 1.
  It is clear by the construction in proof of Theorem 4 that this is possible to achieve while keeping the safety distance. We let $k=an+b$, $a\in \mathbb{N}$, $b\in {\mathbb{N}}_0$.
  If $a\ge 2$, there is still an $M{P}_n^n$ subgraph of $M{P}_n^k$ whose whole set of vertices is unvisited. So from vertices $(1,1)$ and $(n,n)$, f and g move up simultaneously by n steps, thus coming to the corner vertices of such a subgraph. They proceed to visit all the vertices of this subgraph in the same fashion as $f_1$ and $g_1$, at the end coming to the starting corner vertices and then moving up again by n steps if there is still an $M{P}_n^n$ subgraph whose whole set of vertices is unvisited.
  If that is not the case (after a visits $M{P}_n^n$ subgraphs), they both move up by $k-na=b$ steps (the second grid in Figure 12). With this movement, g comes to vertex $(n,k)$ and f comes to vertex $(1,1+k-n)$. They are again in the corner vertices of a subgraph isomorphic to $M{P}_n^n$, so they proceed as before (the third grid in Figure 12).
• n is odd. For odd n, ${\sigma }_V^{\times }\left(M{P}_n^n\right)=n-1$, so we do not use the same construction as in Theorem 4, although our approach is similar. We distinguish two subcases, depending on the parity of k.

  2.1.

  For even k, we can use an analogous construction as in the case $n=k$. Let us just reiterate some key points. We let $f\left(1\right)=(1,1)$ and $g\left(1\right)=(n,k)$. Now, while g alternates between $(n,k)$ and $(n-1,k)$, f moves through all the vertices of subgraph $M{P}_{⌈\frac{n}{2}⌉}^{\frac{k}{2}}$ (the first grid in Figure 13).

  Figure 13. Example of movement for f and g in $M{P}_3^6$ for Case 2.1.

  Figure 13. Example of movement for f and g in $M{P}_3^6$ for Case 2.1.
  Note that when the distance between vertices $(1,1)$ and $\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)$ is even, as in the proof of Theorem 4, f will be in vertex $\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)$ exactly when g is in $(n,k)$ and the distance between those two vertices is $n-⌈\frac{n}{2}⌉+k-\frac{k}{2}=n-\frac{n+1}{2}+\frac{k}{2}=⌊\frac{n}{2}⌋+\frac{k}{2}\ge n$. On the other hand, when the distance between vertices $(1,1)$ and $\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)$ is odd, f will be in vertex $\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)$ when g is in $(n-1,k)$. However, the distance between those two vertices is also at least n. Let us show this. It holds that

  $$d\left((1,1),\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)\right)={\displaystyle \frac{n+1}{2}}-1+{\displaystyle \frac{k}{2}}-1={\displaystyle \frac{n+k-3}{2}}.$$
  Since $n<k$, n is odd and k is even, we have $k\ge n+1$. However, if $k=n+1$, then the distance above equals to $n-1$, which is an even number, and therefore impossible. We conclude that in this case, $k\ge n+3$. Now, let us calculate the distance between $\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)$ and $(n-1,k)$. It holds that

  $$d\left((n-1,k),\left(⌈\frac{n}{2}⌉,\frac{k}{2}\right)\right)=k-{\displaystyle \frac{k}{2}}+n-1-{\displaystyle \frac{n+1}{2}}={\displaystyle \frac{k}{2}}+{\displaystyle \frac{n-3}{2}}\ge {\displaystyle \frac{n+3}{2}}+{\displaystyle \frac{n-3}{2}}=n.$$
  We proceed with f moving right, through path $(1,1),...,(n,1)$ and g moving left, through $(n,k),...,(1,k)$ (the second grid in Figure 13). This movement takes place on the smallest distance f and g reach when they are in the same vertical path, and that distance is $diam{P}_k=k-1\ge n$. They then continue in the same fashion, g alternates between $(1,k)$ and any of its adjacent vertices while f visits the lower right subgraph isomorphic to $M{P}_{⌈\frac{n}{2}⌉}^{⌈\frac{k}{2}⌉}$ (the third grid in Figure 13). In the next stage, f moves up from corner $(n,1)$ to $(n,k)$ while g moves down, from $(1,k)$ to $(1,1)$ (the fourth grid in Figure 13). During this stage, by Lemma 8, f and g are in the same or in the adjacent layers. But, since k is even and they are following active movement rules, it is easy to see that they will never be in the same layer, thus keeping their distance at at least $diam{P}_n+1=n$. The rest of the construction proceeds analogously to that described in Theorem 4.

  2.2.

  For odd k, the construction is not the same, because in the stage while f is moving up on the right vertical brim and g is moving down on the left vertical brim, they will come to the same layer if they both started in the corner vertices. We can remedy that by having g start the movement in vertex $(n-1,k)$ while f starts in $(1,1)$.

  Note that this does not interfere with distances kept in the first stage of the movement, because f will again be in vertex $\left(⌈\frac{n}{2}⌉,⌈\frac{k}{2}⌉\right)$ when g is in corner vertex $(n,k)$ due to the parity of n and k (in the odd number of steps). This stage is shown in the first grid in Figure 14. The distance of f and g, while in the second stage (f moves right horizontally and g moves left horizontally, from $(n-1,k)$ to $(1,k)$, and then down to $(1,k-1)$), is once again at least $diam{P}_k\ge n$ (the second grid in Figure 14). After the next stage, in which f visits the lower right subgraph, for vertical switching, f moves through path $(n,1),..,(n,k)$ and g moves through $(1,k-1),(1,k-2),...,(1,1)$ and then $(2,1)$ (the third grid in Figure 14). In this way, f and g are never in the same layer and their distance is at least $diam{P}_n+1=n$. The rest of the construction once again follows analogously.

  Figure 14. Example of movement for f and g in $M{P}_5^7$ for Case 2.2.

  Figure 14. Example of movement for f and g in $M{P}_5^7$ for Case 2.2.

We prove that ${\sigma }_V^{\times }\left(M{P}_n^k\right)=min\{n,k\}=n$, and since from Theorem 3 we have ${\sigma }_V^{\square }\left(M{P}_n^k\right)=min\{n,k\}-1=n-1$, from result ${\sigma }_V^{⊠}\left(G\right)\ge max\{{\sigma }_V^{\square }\left(G\right),{\sigma }_V^{\times }\left(G\right)\}$ in [3], it follows that ${\sigma }_V^{⊠}\left(M{P}_n^k\right)=n$. □

5. Summary and Conclusions

The summary of our results for multilayered cycles and multilayered paths is given in

Table 1. Vertex span values for $M{C}_n^k$.

${\mathit{\sigma }}_{\mathit{V}}^{⊠}\left({\mathit{MC}}_{\mathit{n}}^{\mathit{k}}\right)$	${\mathit{\sigma }}_{\mathit{V}}^{\times }\left({\mathit{MC}}_{\mathit{n}}^{\mathit{k}}\right)$	${\mathit{\sigma }}_{\mathit{V}}^{\square }\left({\mathit{MC}}_{\mathit{n}}^{\mathit{k}}\right)$
$⌊\frac{n}{2}⌋+1$	$⌊\frac{n}{2}⌋+1$	$⌊\frac{n}{2}⌋$

and

Table 2. Vertex span values for $M{P}_n^k$.

	${\mathit{\sigma }}_{\mathit{V}}^{⊠}\left({\mathit{MP}}_{\mathit{n}}^{\mathit{k}}\right)$	${\mathit{\sigma }}_{\mathit{V}}^{\times }\left({\mathit{MP}}_{\mathit{n}}^{\mathit{k}}\right)$	${\mathit{\sigma }}_{\mathit{V}}^{\square }\left({\mathit{MP}}_{\mathit{n}}^{\mathit{k}}\right)$
$n=k$	$2⌊\frac{n}{2}⌋$	$2⌊\frac{n}{2}⌋$	$n-1$
$n\ne k$	$min\{n,k\}$	$min\{n,k\}$	$min\{n,k\}-1$

.

In conclusion, we see that all vertex spans of multilayered cycles $M{C}_n^k$ depend only on the size of the cycle, n, and not on the number of layers, k. For multilayered paths $M{P}_n^k$, span values depend on the lesser of the two values, n and k. It is worth noting that for $min\{n,k\}=n$, ${\sigma }_V^{\times }\left(M{P}_n^k\right)={\sigma }_V^{\times }\left(M{P}_n^n\right)=n$ for even n, while for odd n, $n={\sigma }_V^{\times }\left(M{P}_n^k\right)>{\sigma }_V^{\times }\left(M{P}_n^n\right)=n-1$.

6. Further Work

Further work on multilayered graphs might include exploring the relation between ${\sigma }_V^{\square }\left(G\right)$, ${\sigma }_V^{\times }\left(G\right)$, ${\sigma }_V^{⊠}\left(G\right)$ and ${\sigma }_V^{\square }\left(M{G}^k\right)$, ${\sigma }_V^{\times }\left(M{G}^k\right)$, ${\sigma }_V^{⊠}\left(M{G}^k\right)$. We are also interested in finding the edge spans for these graph classes. An alternative approach involves extending graph spans to accommodate more than two players traversing the graph, aiming to calculate the optimal safe distance they can maintain. This extension would consider three distinct movement rules: active, where all players move simultaneously; lazy, in which only one player moves at a time while others remain stationary; and traditional, allowing for players to move independently. We believe that this generalization of spans would prove particularly practical for these two graph classes, considering their importance in modeling shopping malls and city streets. As stated in the introduction, it would be interesting to analyze the other aspects of games for which a multilayered cycle or a multilayered path is the playground. From the theoretical point of view, given that multilayered cycle and multilayered path are Cartesian product graphs of a path and a cycle and two paths, respectively, it might be interesting to explore span values for the Cartesian product of two cycle graphs.