Forbidden Pairs of Disconnected Graphs for Traceability of Block-Chains

Abstract

Each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. It occurs to us that forbidden subgraph conditions for a block-chain are traceable. In this article, we characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected).

1. Introduction

The Hamiltonicity of graphs is a very important and far-reaching research topic in structural graph theory. The origin and development of this problem is closely related to the famous Four-Color Conjecture. Therefore, it has attracted the attention of many experts and scholars at home and abroad. The traceability of graphs is also closely related to the research of Hamiltonicity in structural graph theory. In terms of algorithmic complexity, it is NP-complete to determine whether a graph is traceable.

In the paper, we only discuss the finite simple graphs. The undefined terminology and notations can be found in Bondy and Murty [1].

Let $G=\left(V\right(G),E(G\left)\right)$ be a connected graph. We denote the size, order, connectivity, independence number, and component number of G by $e\left(G\right),n\left(G\right),\kappa \left(G\right),\alpha \left(G\right)$, and $\omega \left(G\right)$, respectively. Let u be a vertex of G. The set of vertices that are the neighbors of u in the graph G can be denoted by $N_G\left(u\right)$. Let S be a subset of $V\left(G\right)$(or $E\left(G\right)$). We denote the induced subgraph of G by $G\left[S\right]$. Moreover, we use $G-S$ to denote the subgraph $G\left[V\right(G)\backslash S]$, respectively. For a positive integer l, we define $V_l\left(G\right)=\{v\in V\left(G\right)|d_G\left(v\right)=l\}$.

If a graph includes a Hamilton cycle, then it is said to be to be Hamiltonian, and if it includes a Hamilton path, then it is said to be traceable. Sufficient conditions for the traceability of graphs mainly include the following two categories: one is described from the perspective of parameters, among which the independence number, degree sum, minimum degree, and neighborhood union are commonly used; the other is forbidden certain subgraphs from the perspective of structural graph theory.

Let H be a given graph. If G does not include induced copies of H, then a graph of G is called H-free, and H is said to be a forbidden subgraph of G. For a class of graphs $\mathcal{H}$, if G is H-free for each $H\in \mathcal{H}$, then the graph G is $\mathcal{H}$-free. It should be mentioned that each $H_1$-free graph is also $H_2$-free if $H_1$ is an induced subgraph of $H_2$.

As always, $K_n$ is used to denote the complete graph of order n, and $K_{m,n}$ is used to denote the complete bipartite graph such that partition sets are of size m and n. As a result, a vertex is denoted by $K_1$, a triangle is denoted by $K_3$, and a star is denoted by $K_{1,r}$ (moreover, the $K_{1,3}$ is said to be a claw). Let $K_{2,t}(u,u^{\prime })$ be a $K_{2,t}$ satisfying $u,u^{\prime }$, being the nonadjacent vertices of degree t. Let $K_{2,t}^{\prime }(u,u^{\prime },u^{″})$ be the graph acquired from a $K_{2,t}(u,u^{\prime })$ by adding a new vertex $u^{″}$ that is adjacent to $u^{\prime }$ only. Let $S(m,l)$ be the graph acquired from a $K_{2,m}(u,u^{\prime })$ and a $K_{2,l}^{\prime }(w,w^{\prime },w^{″})$ by identifying u with w and $w^{″}$ with $u^{\prime }$ (see Figure 1).

Here are some graphs that will be used later (see Figure 2):

• $P_i$, the path having i vertices (it should be noted that $P_1=K_1$ and $P_2=K_2$);
• $Z_i$, a graph acquired by identifying an end-vertex of a $P_{i+1}$ with a vertex of a $K_3$;
• $B_{i,j}$, a graph acquired by identifying an end-vertex of a $P_{i+1}$ and $P_{j+1}$ with the two vertices of a $K_3$, respectively;
• $N_{i,j,k}$, a graph acquired by identifying an end-vertex of a $P_{i+1},P_{j+1}$, and $P_{k+1}$ with the three vertices of a $K_3$, respectively;
• $T_{i,j,k}\left(v\right)$, a graph acquired by identifying an end-vertex of three paths $P_{i+1},P_{j+1}$, and $P_{k+1}$ with one vertex v, respectively; moreover, when there is no scope for confusion, we use the notation $T_{i,j,k}$ to denote $T_{i,j,k}\left(v\right)$;
• $L_i$, a graph acquired by identifying an end-vertex of a $P_2$ with a vertex of a $K_i$.

A graph is an empty graph (no edge in the graph), if it is $P_2$-free. A graph is a clique if it is $2K_1$-free. It is not too difficult to check that the graph is traceable. For fear of this trivial case, we just consider that our forbidden subgraphs have at least three vertices throughout the whole article.

Bedrossian [2] characterized all pairs $\{R,S\}$ of connected forbidden subgraphs for Hamiltonicity. Chen et al. [3] considered pairs of forbidden subgraphs that force Hamiltonicity in a graph of high connectivity. Xiong et al. [4] considered pairs of forbidden subgraphs for the edge-connectivity of a connected graph. The most representative result of a traceable graph obtained by adding the forbidden subgraph condition is the conclusion about pairs of forbidden subgraphs [5].

Faudree and Gould [5] extended Bedrossian’s result and characterized all the pairs $\{R,S\}$ of connected subgraphs for traceability.

Theorem 1

([5]). Let R and S be connected graphs such that $R,S\ne P_3$, and let G be a connected graph. Then, G with $\{R,S\}$-free indicates G is traceable if and only if (up to symmetry) $R=K_{1,3}$ and S is an induced subgraph of $N_{1,1,1}$.

With more and more in-depth research on forbidden subgraphs, most studies focus on connected forbidden subgraphs, while the research on disconnected forbidden subgraphs is very few and just started. The first people who studied disconnected forbidden subgraphs were Li and Vrána [6]. They come up with the disconnected forbidden subgraphs and characterized all pairs of disconnected forbidden subgraphs for Hamiltonicity.

Theorem 2

([6]). Let S be a graph with $\left|V\right(S\left)\right|\ge 3$. Then, every two-connected S-free graph is Hamiltonian if and only if S is $P_3$ or $3K_1$.

Theorem 3

([6]). Let R and S be graphs of order at least three other than $P_3$ and $3K_1$. Then, there exists an integer $n_0$ satisfying every two-connected $\{R,S\}$-free graph of order at least $n_0$ that is Hamiltonian, if and only if one of the following is true (up to symmetry):

• $R=K_{1,3}$ and S is an induced subgraph of $P_6,Z_3,N_{0,1,2},N_{1,1,1},K_1\bigcup Z_2,K_2\cup Z_1$, or $K_3\cup P_4$;
• $R=K_{1,k}$ with $k\ge 4$ and S is an induced subgraph of $2K_1\cup K_2$;
• $R=k{K}_1$ with $k\ge 4$ and S is an induced subgraph of $L_l$ with $l\ge 3$, or $2K_1\cup K_{l^{\prime }}$ with $l^{\prime }\ge 2$.

So far, more and more scholars have begun to study pairs of disconnected forbidden subgraphs and many results have been obtained. Xiong et al. [7] considered the pairs of disconnected forbidden subgraphs for a two-factor connected graph. In [8], Du, Li and Xiong extended the result made by Faudree and Gould [5] and characterized all pairs of disconnected forbidden subgraphs for traceability.

Theorem 4

([8]). Let S be a graph with $\left|V\right(S\left)\right|\ge 3$. Then, every connected S-free graph is traceable if and only if S is $P_3$ or $3K_1$.

Theorem 5

([8]). Let R and S be graphs of order at least three other than $P_3$ and $3K_1$. Then, there exists an integer $n_0$ satisfying every connected $\{R,S\}$-free graph of order at least $n_0$ where is traceable, if and only if one of the following is true (up to symmetry):

• $R=K_{1,3}$ and S is an induced subgraph of $N_{1,1,1},K_1\cup Z_1$, or $2K_1\cup K_3$;
• $R=K_{1,k}$ with $k\ge 4$ and S is an induced subgraph of $2K_1\cup K_2$;
• $R=k{K}_1$ with $k\ge 4$ and S is an induced subgraph of $L_l$ with $l\ge 3$, or $2K_1\cup K_{l^{\prime }}$ with $l^{\prime }\ge 2$.

Let G be a graph. If it does not have separating vertices and is connected, then G is called nonseparable. We define a maximal nonseparable subgraph ($P_1$, or $P_2$, or two-connected) of G as a block of G. We use end-block to denote the block that includes exactly one cut vertex of G. If a graph has connectivity 1 and has exactly two end-blocks or is nonseparable, then it is called a block-chain. We should pay attention that each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. Many graphs employed in the “only if” part of the proof of Theorem 1 are not block-chains (and are therefore trivially non-traceable). It occurs to us that forbidden subgraph conditions for a block-chain are traceable. In [9], Li, Broersma, and Zhang characterized all pairs of connected graphs R and S other than $P_3$, ensuring that every $\{R,S\}$-free block-chain is traceable.

Theorem 6

([9]). The only connected graph S satisfying a block-chain with S-free indicates it is traceable $P_3$.

Theorem 7

([9]). Let R and S be connected graphs such that R, $S\ne P_3$. Then, a block-chain with $\{R,S\}$-free indicates it is traceable if and only if (up to symmetry) $R=K_{1,3}$ and S is an induced subgraph of $N_{1,1,3}$, or $R=K_{1,4}$ and $S=P_4$.

It is natural that we consider pairs of disconnected forbidden subgraphs for the traceability of block-chains.

We define the Ramsey number $r(k,l)$ as the smallest integer satisfying that each graph of order $r(k,l)$ includes either a clique of k vertices or an independent set of l vertices [10].

In [11], Lei et al. characterized all pairs of disconnected forbidden subgraphs excepting a claw for the traceability of block-chains.

Theorem 8

([11]). Let S be a graph with $\left|V\right(S\left)\right|\ge 3$. Then, every S-free block-chain of order at least $r(10,4)$ is traceable if and only if S is $P_3$ or $4K_1$.

Theorem 9

([11]). Let R and S be graphs of order at least three other than $P_3$, $4K_1$, and $K_{1,3}$. Then, there exists an integer $n_0=max\{r(10,4),r(3k+5,k+2),r(2l-3,k)+k-2,r(2k+l^{\prime }-2,k)\}$ satisfying every $\{R,S\}$-free block-chain of order at least $n_0$ being traceable, if and only if one of the following is true (up to symmetry):

• $R=K_{1,4}$ and S is an induced subgraph of $P_4,P_3\cup K_1$, or $3K_1\cup K_2$;
• $R=K_{1,k}$ with $k\ge 5$ and S is an induced subgraph of $3K_1\cup K_2$;
• $R=k{K}_1$ with $k\ge 5$ and S is an induced subgraph of $L_l$ with $l\ge 3$, or $3K_1\cup K_{l^{\prime }}$ with $l^{\prime }\ge 2$.

Clearly, Lei et al. [11] did not characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains. The authors did not consider the case where one of the forbidden subgraphs in pairs of forbidden subgraphs is a claw. Li et al. [9] characterized all pairs of connected forbidden subgraphs for the traceability of block-chains. They did not consider disconnected forbidden subgraphs for the traceability of block-chains. In this paper, we will characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected).

In the following sections, Section 1 is devoted to the main results and discussions in this paper, Section 2 is devoted to the conclusions, and gives possible avenues for future work, Appendix A.1 is devoted to the preliminaries for the proof of Theorems 10–12, Appendix A.2 is devoted to the proofs of Theorem 10, and Appendix A.3 is devoted to the proofs of Theorem 11.

2. Results and Discussions

In this paper, we will characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains.

We construct families of block-chains ${\mathcal{G}}_0$ that are not traceable. ${\mathcal{G}}_0$ is a family of graphs H obtained from six complete graphs $B_i$ for $i\in \{1,\dots ,6\}$ with $V\left(B_1\right)\cap V\left(B_2\right)=\left\{v_1\right\}$, $V\left(B_2\right)\cap V\left(B_3\right)=\left\{v_2\right\}$, $V\left(B_2\right)\cap V\left(B_4\right)=\left\{v_3\right\}$, $V\left(B_3\right)\cap V\left(B_5\right)=\left\{v_4\right\}$, $V\left(B_4\right)\cap V\left(B_5\right)=\left\{v_5\right\}$, $V\left(B_5\right)\cap V\left(B_6\right)=\left\{v_6\right\}$, and $\left|V\right(B_j\left)\right|\ge 2$ for $j\in \{1,6\}$, $\left|V\right(B_k\left)\right|\ge 3$ for $k\in \{2,3,4\}$, $\left|V\right(B_5\left)\right|=3$. We also construct families of graph ${\mathcal{F}}_0$ such that $L\left({\mathcal{F}}_0\right)={\mathcal{G}}_0$; see Figure 3 (where the edge that is possibly zero is shown as a dotted line).

We will characterize all pairs of disconnected $\{R,S\}$ of graphs such that there is an integer $n_0$ satisfying every disconnected $\{R,S\}$-free block-chain of order at least $n_0$ being traceable by the following theorems.

Theorem 10.

Let G be a $K_{1,3}$-free block-chain. Then:

(1) 

If G is also $P_6\cup K_1$-free, then G is traceable;

(2) 

If G is also $Z_2\cup 2K_1$-free and $n\left(G\right)\ge 183$, then G is traceable;

(3) 

If G is also $Z_3\cup K_1$-free and $n\left(G\right)\ge 32$, then G is traceable;

(4) 

If G is also $K_3\cup P_4\cup K_1$-free and $n\left(G\right)\ge 3482$, then G is traceable;

(5) 

If G is also $N_{1,1,1}\cup K_2$-free, then G is traceable;

(6) 

If G is also $N_{1,1,2}\cup K_1$-free, then either G is traceable or G is isomorphic to a member of ${\mathcal{G}}_0$.

See Appendix A.2 for the proof of Theorem 10.

Corollary 1.

If G is a $\{K_{1,3},B_{1,2}\cup K_1\}$-free block-chain, then G is traceable.

Proof of Corollary 1.

By contradiction, suppose that G is a $\{K_{1,3},B_{1,2}\cup K_1\}$-free non-traceable block-chain. Note that G is $B_{1,2}\cup K_1$-free, then G must be $N_{1,1,2}\cup K_1$-free. By Theorem 10 (6), the closure $cl\left(G\right)$ of G is isomorphic to a member of ${\mathcal{G}}_0$. Note that $B_i$ is a clique in $cl\left(G\right)$; possibly, $G\left[V\left(B_i\right)\right]$ is not a clique. However, $G\left[V\left(B_i\right)\right]$ must be connected. Moreover, if $\left|V\right(B_i\left)\right|\ge 3$, then $G\left[V\left(B_i\right)\right]$ must be two-connected.

Then, for $\{v_2,v_4\}\in V\left(B_3\right)$ (see Figure 3), there exists a shortest path joining $v_2$ and $v_4$ in $G\left[V\left(B_3\right)\right]$. The length of a shortest such path is called the distance of $G\left[V\left(B_3\right)\right]$ between $v_2$ and $v_4$ and denoted $d_{G\left[V\left(B_3\right)\right]}(v_2,v_4)$. We shall distinguish the following cases to prove Corollary 1.

Case 1$d_{G\left[V\left(B_3\right)\right]}(v_2,v_4)=1$.

Note that $\left|V\right(B_3\left)\right|\ge 3$. Then, there is at least one vertex $x_1\in V\left(B_3\right)$ with $N_{G\left[V\left(B_3\right)\right]}\left(x_1\right)\cap \{v_2,v_4\}\ne \varnothing$. Since G is $K_{1,3}$-free, $G\left[\{v_2,v_4,x_1\}\right]$ is a clique. By the definition of ${\mathcal{F}}_0$, there exist two vertices $\{x_2,x_3\}\subseteq V(B_1\cup B_2)$ such that $v_2{x}_2{x}_3$ is a path of G. Then, $G\left[\{x_1,v_2,v_4,v_6,x_2,x_3\}\right]\cong B_{1,2}$. There exists a vertex $x_4\in V\left(B_4\right)$ with $N_G\left(x_4\right)\cap V\left(G\left[\{x_1,v_2,v_4,v_6,x_2,x_3\}\right]\right)=\varnothing$. Then, $G\left[\{x_1,v_2,v_4,v_6,x_2,x_3,x_4\}\right]\cong B_{1,2}\cup K_1$, a contradiction.

Case 2$d_{G\left[V\left(B_3\right)\right]}(v_2,v_4)>1$.

There exists at least two vertices $\{y_1,y_2\}\subseteq V\left(B_3\right)$ such that $v_4{y}_1{y}_2$ is a path of $G\left[V\left(B_3\right)\right]$.There exists a vertex $y_3\in V\left(B_6\right)$ with $v_6{y}_3\in E\left(G\right)$. Then, $G\left[\{v_4,v_5,v_6,y_1,y_2,y_3\}\right]\cong B_{1,2}$. There exists a vertex $y_4\in V\left(B_1\right)$ with $N_G\left(y_4\right)\cap V\left(G\left[\{v_4,v_5,v_6,y_1,y_2,y_3\}\right]\right)=\varnothing$. Then, $G\left[\{v_4,v_5,v_6,y_1,y_2,y_3,y_4\}\right]\cong B_{1,2}\cup K_1$, a contradiction.

These contradictions show that Corollary 1 holds. □

Theorem 11.

Let G be a $K_{1,3}$-free block-chain. Then, if G is also $S\cup P_4$-free for $S\in \{Z_1,P_4\}$ and $n\left(G\right)\ge 212$, then G is traceable.

See Appendix A.3 for the proof of Theorem 11.

Ultimately, we may propose our main result as below.

Theorem 12.

Let R and S be graphs of order at least three other than $P_3$ and $4K_1$. Then, there is an integer $n_0$ such that every $\{R,S\}$-free block-chain of order at least $n_0$ is traceable, if and only if one of the following is true (up to symmetry):

• $R=K_{1,3}$ and S is an induced subgraph of $N_{1,1,3},P_6\cup K_1,Z_2\cup 2K_1,$$Z_3\cup K_1,K_3\cup P_4\cup K_1,Z_1\cup P_4,P_4\cup P_4,N_{1,1,1}\cup K_2$, or $B_{1,2}\cup K_1$;
• $R=K_{1,4}$ and S is an induced subgraph of $P_4,P_3\cup K_1$ or $3K_1\cup K_2$;
• $R=K_{1,k}$ with $k\ge 5$ and S is an induced subgraph of $3K_1\cup K_2$;
• $R=k{K}_1$ with $k\ge 5$ and S is an induced subgraph of $L_l$ with $l\ge 3$, or $3K_1\cup K_{l^{\prime }}$ with $l^{\prime }\ge 2$.

Proof of Theorem 12.

By Theorem 1 and Theorems 9–11, the sufficiency is proven. Then, we only need to prove the necessity.

Firstly, some families of block-chains ${\mathcal{G}}_i,i=1,\dots ,6$, which are not traceable (see Figure 4), are constructed.

Let R and S be graphs of order at least three other than $P_3$ and $4K_1$, such that there exists an integer $n_0$ such that every connected $\{R,S\}$-free block-chain of order at least $n_0$ is traceable. Then, all graphs in ${\mathcal{G}}_i,i=1,\dots ,6$, of order large enough, contain either R or S as an induced subgraph. By Theorem 9, it suffices to consider that either R or S is $K_{1,3}$. We can assume without loss of generality that $R=K_{1,3}$. Note that all graphs in ${\mathcal{G}}_1,{\mathcal{G}}_2,{\mathcal{G}}_3,{\mathcal{G}}_4,{\mathcal{G}}_5,{\mathcal{G}}_6$ are $K_{1,3}$-free. Thus, each of them contains S being an induced subgraph. Since all graphs in ${\mathcal{G}}_1$ are $5K_1$-free, S is also $5K_1$-free, i.e., $\alpha \left(S\right)\le 4$. This indicates that $\omega \left(S\right)\le 4$. If $\omega \left(S\right)=1$, i.e., S is connected, then, by Theorem 7, S is an induced subgraph of $N_{1,1,3}$. Therefore, we assume that $\omega \left(S\right)>1$, i.e., S is disconnected. Note that S is an induced subgraph of all graphs in ${\mathcal{G}}_2$, and S is also an induced subgraph of all graphs in ${\mathcal{G}}_3$. Then, if there exists one component of S such that it contains a cycle, then the cycle is triangle. If every component of S does not contain a triangle, then S is a forest. Since G is $K_{1,3}$-free, every component of S is a path. Note that all graphs in ${\mathcal{G}}_4$ are $K_3\cup K_3$-free. Therefore, if there exists one component of S such that it contains a triangle, say $H_i$ for $1\le i\le 4$, then every component of $S-H_i$ is a path. Suppose that $\omega \left(S\right)=2$, say, $H_1,H_2$ are the two components of S. Suppose first that $H_1$ and $H_2$ are paths of G. Observe that all graphs in ${\mathcal{G}}_5$ are $P_7$-free. Then, $H_1$ is an induced subgraph of $P_6$. If $H_1=K_1$, then noting that all graphs in ${\mathcal{G}}_5$ are $K_1\cup P_7$-free, we have that $H_2$ is an induced subgraph of $P_6$; if $H_1=K_2$, then observing that all graphs in ${\mathcal{G}}_5$ are $K_2\cup P_5$-free, we have that $H_2$ is an induced subgraph of $P_4$; if $H_1=P_3$, then observing that all graphs in ${\mathcal{G}}_5$ are $P_3\cup P_5$-free, we have that $H_2$ is an induced subgraph of $P_4$; if $H_1=P_4$; then observing that all graphs in ${\mathcal{G}}_5$ are $P_4\cup P_5$-free, we have that $H_2$ is an induced subgraph of $P_4$; if $H_1=P_5$; then observing that all graphs in ${\mathcal{G}}_5$ are $P_5\cup K_2$-free, we have that $H_2$ is an induced subgraph of $K_1$; if $H_1=P_6$, then observing that all graphs in ${\mathcal{G}}_5$ are $P_6\cup K_2$-free, we have that $H_2$ is an induced subgraph of $K_1$. Furthermore, note that $P_5\cup K_1$ is an induced subgraph of $P_6\cup K_1$ and $P_4\cup P_3$ is an induced subgraph of $P_4\cup P_4$. Thus, S is an induced subgraph of $P_6\cup K_1$ or $P_4\cup P_4$. Suppose now that $H_1$ contains a triangle and $H_2$ is a path of G. Then, $H_1$ is an induced subgraph of $K_3,Z_i,B_{i,j},N_{i,j,k}$. If $H_1$ is an induced subgraph of $N_{i,j,k}$, then, by $\alpha \left(S\right)\le 4$, $H_2$ is $K_1$ or $K_2$. Note that all graphs in ${\mathcal{G}}_1$ are $\{N_{1,1,2}\cup K_1,N_{1,1,1}\cup P_3\}$-free. Thus, S is an induced subgraph of $N_{1,1,1}\cup K_2$. If $H_1$ is an induced subgraph of $B_{i,j}$, then, by $\alpha \left(S\right)\le 4$, $H_2$ is $K_1$ or $K_2$. All graphs in ${\mathcal{G}}_5$ are $\{B_{1,2}\cup K_2\}$-free, and all graphs in ${\mathcal{G}}_6$ are $\{B_{2,2}\cup K_1\}$-free. Thus, S is an induced subgraph of $B_{1,2}\cup K_1$. If $H_1$ is an induced subgraph of $Z_i$, then noting that all graphs in ${\mathcal{G}}_5$ are $\{Z_3\cup K_2,Z_2\cup K_2,Z_1\cup P_5\}$-free, also note that $Z_2\cup K_1$ is an induced subgraph of $Z_3\cup K_1$. Thus, S is an induced subgraph of $Z_3\cup K_1$ or $Z_1\cup P_4$. If $H_1$ is $K_3$, then note that all graphs in ${\mathcal{G}}_5$ are $K_3\cup P_5$-free. Thus, $H_2$ is an induced subgraph of $K_3\cup P_4$. Suppose that $\omega \left(S\right)=3$, say $H_1,H_2$, and $H_3$ are three components of S. By $\omega \left(S\right)\le 4$, up to symmetry, $H_1$ and $H_2$ are complete and $\omega \left(S\right)\le 2$. This implies that $H_3$ is either an induced subgraph of $Z_2$ or an induced subgraph of $P_6$. Note that all graphs in ${\mathcal{G}}_5$ are $\{Z_2\cup K_1\cup K_2,Z_1\cup K_2\cup K_2,K_3\cup P_5\cup K_1\}$-free, and also observe that $K_3\cup P_4$ is an induced subgraph of $K_3\cup P_4\cup K_1$. Thus, S is an induced subgraph of $Z_2\cup 2K_1$ or $K_3\cup P_4\cup K_1$. Suppose that $\omega \left(S\right)=4$. Then, every component of S is complete and of order at most three. Note that all graphs in ${\mathcal{G}}_5$ are $K_3\cup 2K_2\cup K_1$-free; also note that $K_3\cup K_2\cup 2K_1$ is an induced subgraph of $K_3\cup P_4\cup K_1$. We conclude that S is an induced subgraph of $K_3\cup P_4\cup K_1$. This completes the proof of the necessity. □

Concluding remark: In this paper, we characterized all pairs $\{R,S\}$ of graphs (not necessary connected) such that there exists an integer $n_0$ such that every connected $\{R,S\}$-free graph of order at least $n_0$ is traceable.

3. Conclusions

In 2013, Li, Broersma, and Zhang [9] characterized all the pairs of connected forbidden subgraphs for the traceability of block-chains. We characterized all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected). This further reveals the profound connotation of graph traceability and the Hamiltonicity property.

In the future, we can consider the triples’ disconnected forbidden subgraphs for traceability and Hamiltonicity. Forbidden subgraphs are closely related to many path and cycle properties of graphs, which have attracted the attention of many graph theory experts. Many meaningful research results have been obtained, and many problems worth further study have been left. The study of disconnected forbidden subgraphs is just beginning, and the properties of many graphs are related to disconnected forbidden subgraphs. In terms of disconnected forbidden subgraphs and pancyclicity, the disconnected forbidden subgraphs and Hamiltonian-connected onescan be further studied. With the application of disconnected forbidden subgraphs, people can also study the color, matching, factor, and other problems of the graph. The research prospect of disconnected forbidden subgraphs is very broad, and there are still many problems to break through.