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17 pages, 1696 KiB

Open AccessArticle

The Edge Odd Graceful Labeling of Water Wheel Graphs

by Mohammed Aljohani and Salama Nagy Daoud

Axioms 2025, 14(1), 5; https://doi.org/10.3390/axioms14010005 - 26 Dec 2024

Viewed by 925

A graph,

G = (V, E)

, is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection [...] Read more.

A graph,

G = (V, E)

, is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection

g : E (G) \to {1, 3, \dots, 2 m - 1}

, from which an injective transformation is derived,

g^{*} : V (G) \to {1, 2, 3, \dots, 2 m - 1}

, from the rule that the image of

u \in V (G)

under

g^{*}

\sum_{u v \in E (G)} g (u v) \mod (2 m)

. The main objective of this manuscript is to introduce new classes of planar graphs, namely water wheel graphs,

W W_{n}

; triangulated water wheel graphs,

T W_{n}

; closed water wheel graphs,

C W_{n}

; and closed triangulated water wheel graphs,

C T_{n}

. Furthermore, we specify conditions for these graphs to allow for edge odd graceful labelings. Full article

(This article belongs to the Section Algebra and Number Theory)

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10 pages, 510 KiB

Open AccessArticle

A Novel View of Closed Graph Function in Nano Topological Space

by Kiruthika Kittusamy, Nagaveni Narayanan, Sheeba Devaraj and Sathya Priya Sankar

Axioms 2024, 13(4), 270; https://doi.org/10.3390/axioms13040270 - 18 Apr 2024

Viewed by 1431

Abstract

The objective of this research is to describe and investigate a novel class of separation axioms and discuss some of their fundamental characteristics using a nano weakly generalized closed set. In nano topological space,

N w g

-closed graph and strongly [...] Read more.

N w g

-closed graph and strongly

N w g

-closed graph functions are introduced and explored. We also analyse some of the characterizations of closed graph functions with the separation axioms via a nano weakly generalized closed set. Full article

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15 pages, 683 KiB

Open AccessArticle

Another Antimagic Conjecture

by Rinovia Simanjuntak, Tamaro Nadeak, Fuad Yasin, Kristiana Wijaya, Nurdin Hinding and Kiki Ariyanti Sugeng

Symmetry 2021, 13(11), 2071; https://doi.org/10.3390/sym13112071 - 2 Nov 2021

Cited by 5 | Viewed by 2658

Abstract

An antimagic labeling of a graph G is a bijection

f : E (G) \to {1, \dots, | E (G) |}

such that the weights [...] Read more.

An antimagic labeling of a graph G is a bijection

f : E (G) \to {1, \dots, | E (G) |}

such that the weights

w (x) = \sum_{y \sim x} f (y)

distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than

K_{2}

admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection

f : V (G) \to {1, \dots, | V (G) |}

such that the weight

ω (x) = \sum_{y \in N_{D} (x)} f (y)

is distinct for each vertex x, where

N_{D} (x) = {y \in V (G) | d (x, y) \in D}

is the D-neigbourhood set of a vertex x. If

N_{D} (x) = r

, for every vertex x in G, a graph G is said to be $(D, r)$ -regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for

D = {1}

, all graphs of order up to 8 concur with the conjecture. We prove that the set of

(D, r)

-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric

(D, r)

-regular that are D-antimagic and examples of disjoint union of non-symmetric non-

(D, r)

-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph. Full article

(This article belongs to the Special Issue Graph Labelings and Their Applications)

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14 pages, 297 KiB

Open AccessArticle

From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

by Abel Cabrera Martínez, Alejandro Estrada-Moreno and Juan Alberto Rodríguez-Velázquez

Symmetry 2021, 13(6), 1036; https://doi.org/10.3390/sym13061036 - 8 Jun 2021

Cited by 4 | Viewed by 2008

Abstract

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex [...] Read more.

x \in V (G)

of a graph G, the neighbourhood of x is denoted by

N (x)

. The neighbourhood of a set

X \subseteq V (G)

is defined to be

N (X) = ⋃_{x \in X} N (x)

, while the external neighbourhood of X is defined to be

N_{e} (X) = N (X) ∖ X

. Now, for every set

X \subseteq V (G)

and every vertex

x \in X

, the external private neighbourhood of x with respect to X is defined as the set

P_{e} (x, X) = {y \in V (G) ∖ X : N (y) \cap X = {x}} .

Let

X_{w} = {x \in X : P_{e} (x, X) \neq ⌀}

. The strong differential of X is defined to be

\partial_{s} (X) = | N_{e} (X) | - | X_{w} |

, while the quasi-total strong differential of G is defined to be

\partial_{s^{*}} (G) = max {\partial_{s} (X) : X \subseteq V (G) and X_{w} \subseteq N (X)} .

We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard. Full article

(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics)

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20 pages, 356 KiB

Open AccessArticle

Total Weak Roman Domination in Graphs

by Abel Cabrera Martínez, Luis P. Montejano and Juan A. Rodríguez-Velázquez

Symmetry 2019, 11(6), 831; https://doi.org/10.3390/sym11060831 - 24 Jun 2019

Cited by 12 | Viewed by 3589

Abstract

Given a graph

G = (V, E)

, a function

f : V \to {0, 1, 2, \dots}

is said to be a total dominating function if [...] Read more.

Given a graph

G = (V, E)

, a function

f : V \to {0, 1, 2, \dots}

is said to be a total dominating function if

\sum_{u \in N (v)} f (u) > 0

for every

v \in V

, where

N (v)

denotes the open neighbourhood of v. Let

V_{i} = {x \in V : f (x) = i}

. We say that a function

f : V \to {0, 1, 2}

is a total weak Roman dominating function if f is a total dominating function and for every vertex

v \in V_{0}

there exists

u \in N (v) \cap (V_{1} \cup V_{2})

such that the function

f^{'}

, defined by

f^{'} (v) = 1

f^{'} (u) = f (u) - 1

and

f^{'} (x) = f (x)

whenever

x \in V {u, v}

, is a total dominating function as well. The weight of a function f is defined to be

w (f) = \sum_{v \in V} f (v) .

In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by

γ_{t r} (G)

, which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on

γ_{t r} (G)

and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard. Full article

(This article belongs to the Section Mathematics)

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