Traceability of connected domination critical graphs

Abstract

A dominating set in a graph G is a set S of vertices of G such that every vertex outside S is adjacent to a vertex in S. A connected dominating set in G is a dominating set S such that the subgraph G[S] induced by S is connected. The connected domination number of G, γ_c(G), is the minimum cardinality of a connected dominating set of G. A graph G is said to be k-γ_c-critical if the connected domination number γ_c(G) is equal to k and ${\gamma }_c(G+uv)<k$ for every pair of non-adjacent vertices u and v of G. Let ζ be the number of cut-vertices of G. It is known that if G is a k-γ_c-critical graph, then G has at most $k-2$ cut-vertices, that is, $\zeta \le k-2$. In this paper, for k ≥ 4 and $0\le \zeta \le k-2,$ we show that every k-γ_c-critical graph with ζ cut-vertices has a hamiltonian path if and only if $k-3\le \zeta \le k-2$.

Introduction

A dominating set in a graph G is a set S of vertices of G such that every vertex in V(G)∖S is adjacent to at least one vertex in S. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. A graph G is said to be k-γ-critical if $\gamma \left(G\right)=k$ and $\gamma (G+uv)<k$ for every pair of non-adjacent vertices u and v of G. Such a graph G is called a domination critical graph. If S is a dominating set of G, we write S≻G, and if $S=\left\{v\right\},$ we also write v≻G rather than {v}≻G. The concept of domination and its variations have been widely studied in the literature; a rough estimate says that it occurs in more than 6000 papers to date. A thorough treatment of the fundamentals of domination theory in graphs can be found in the books [12], [13].

A connected dominating set, abbreviated a CD-set, of a connected graph G is a dominating set S of G such that the subgraph G[S] induced by S is connected. The connected domination number of G, denoted by γ_c(G), is the minimum cardinality of a CD-set of G. A CD-set of G of cardinality γ_c(G) is called a γ_c-set of G. A graph G is said to be k-γ_c-critical if ${\gamma }_c\left(G\right)=k$ and ${\gamma }_c(G+uv)<k$ for every pair of non-adjacent vertices u and v of G. Such a graph G is called a connected domination critical graph. If S is a CD-set of G, we write S≻_cG, and if $S=\left\{v\right\},$ we also write v≻_cG rather than {v}≻_cG. The concept of connected domination was studied at least in the early 1970s, although it was first formally defined by Sampathkumar and Walikar in their 1979 paper [23]. Subsequently over the past forty years, the connected domination number has been extensively studied in the literature; a rough estimate says that it occurs in more than 400 papers to date. For a small sample of papers on the connected domination we refer the reader to [3], [7], [8], [21], [22], [24].

We remark that the concept of connected domination in graphs is application driven, as evidenced by the earlier papers on the concept. For example, Wu and Li [28] showed that connected dominating sets are useful in the computation of routing for mobile ad hoc networks. In this application, a minimum connected dominating set is used as a backbone for communications, and vertices that are not in this set communicate by passing messages through neighbors that are in the set.

We also remark that finding connected dominating sets and Steiner trees in a graph are closely related [5], [6]. Moreover, determining the connected domination number of a connected graph G is equivalent to finding the largest possible number of leaves among all spanning trees of G. A maximum leaf spanning tree of G is a spanning tree that has the largest possible number of leaves among all spanning trees of G, and the max leaf number, denoted by ℓ_max(G), of G is the number of leaves in a maximum leaf spanning tree of G. Since $n\left(G\right)={\ell }_{\max }\left(G\right)+{\gamma }_c\left(G\right),$ the problems of a connected dominating set and a maximum leaf spanning tree are closely connected. The maximum leaf spanning tree problem is MAX-SNP hard, implying that no polynomial time approximation scheme is likely [11]. We remark, however, that both the minimum connected dominating set problem and the maximum leaf spanning tree problem are fixed-parameter tractable [2]. The connected dominating set problem is polynomially solvable for distance-hereditary graphs [6].

For notation and graph theory terminology, we in general follow [14]. Specifically, let $G=(V,E)$ be a graph with vertex set $V=V\left(G\right)$ and edge set $E=E\left(G\right),$ and let v be a vertex in V. A neighbor of a vertex is a vertex adjacent to it. The open neighborhood of v is the set N_G(v) of all neighbors of v, and so $N_G\left(v\right)=\{u\in V|uv\in E\}$ and the closed neighborhood of v is $N_G\left[v\right]=\left\{v\right\}\cup N_G\left(v\right)$. A vertex v is said to dominate a vertex u in G if $u=v$ or if u is a neighbor of v. The degree of a vertex v is |N_G(v)| and is denoted by d_G(v). An end vertex is a vertex of degree 1 and a support vertex is a vertex adjacent to an end vertex. For a set S of vertices in G, the subgraph induced by S in G is denoted by G[S]. If G is a graph, the complement of G, denoted by $\overline{G},$ is formed by taking the vertex set of G and joining two vertices by an edge whenever they are not joined in G. If the graph G is clear from the context, we omit it in the above expressions. For example, we write N(v) and N[v] rather than N_G(v) and N_G[v], respectively. We use the standard notation $\left[k\right]=\{1,\dots ,k\}$.

Two vertices u and v in a graph G are connected if there exists a (u, v)-path in G. A graph G is connected if every two vertices in G are connected. We denote the number of components in a graph G by ω(G). The distance d_G(u, v) between two vertices u and v in a connected graph G is the length of a shortest (u, v)-path in G. A hamiltonian cycle (respectively, hamiltonian path) of a graph is a cycle (path) passing through all vertices of the graph. A graph G is traceable if it contains a hamiltonian path. Moreover, a graph G is hamiltonian if it contains a hamiltonian cycle. For any subgraph F of G and distinct vertices a and b of G, aP_Fb denotes an (a, b)-path in G all of whose internal vertices are in V(F). We note that a and b need not be in V(F). If P is an (a, b)-path in G, we sometimes write the path P by aPb to indicate the start and end vertices of the path P.

We denote the path, cycle, and complete graph on n vertices by P_n, C_n, and K_n, respectively, and we denote the complete bipartite graph with partite sets of cardinality n and m by K_n,m. A star is the graph K_1,k, where k ≥ 1. The graph K_1,3 is called a claw. A graph G is claw-free if it does not contain a claw as an induced subgraph. A tree is a connected graph with no cycle.

For vertex subsets X, Y⊆V(G), we let N_Y(X) be the set of all vertices in Y that have a neighbor that belongs to X in G, that is, $N_Y\left(X\right)=\{y\in Y\mid y\in N_G\left(x\right)$ for some x ∈ X}. For a subgraph H of G, we use N_Y(H) instead of N_Y(V(H)) and we use N_H(X) instead of N_V(H)(X). If $X=\left\{x\right\},$ we use N_Y(x) instead of N_Y({x}). The open neighborhood of a set S of vertices in G is the set $N_G\left(S\right)={\bigcup }_{v\in S}{N}_G\left(v\right)$ and its closed neighborhood is the set $N_G\left[S\right]=N_G\left(S\right)\cup S$.

A subset S⊆V(G) is a vertex cut set of G if the number of components of $G-S$ is more than the number of components of G; that is, $\omega (G-S)>\omega \left(G\right)$. In particular, if $S=\left\{v\right\},$ then v is called a cut-vertex of G. We let ζ(G) be the number of cut-vertices of G. When no ambiguity can occur, we write ζ instead of ζ(G). A block of a graph G is a maximal connected subgraph of G has no cut-vertex of its own. Thus, a block is a maximal 2-connected subgraph of G. Any two blocks of a graph have at most one vertex in common, namely a cut-vertex. A block of G containing exactly one cut-vertex of G is called an end block. If a connected graph contains a single block, we call the graph itself a block.

For ℓ ≥ 2 and a finite sequence $G_1,\cdots ,G_{\ell }$ of vertex disjoint graphs, we let the join G₁∨⋅⋅⋅∨G_ℓ be the graph obtained from the disjoint union of $G_1,\cdots ,G_{\ell }$ by joining each vertex in G_i to all vertices in $G_{i+1}$ for $i\in [\ell -1]$. If $V\left(G_i\right)=\left\{x\right\},$ then we write $G_1\vee \cdots \vee G_{i-1}\vee x\vee G_{i+1}\vee \cdots \vee G_{\ell }$. Moreover, for vertex disjoint graphs G₁ and G₂ and for a subgraph H of G₂, the join G₁∨_HG₂ is the graph obtained from the disjoint union of G₁ and G₂ by joining each vertex in G₁ to each vertex in H.

A study of properties of domination critical graphs was initiated by Sumner and Blitch in their classical 1983 paper [25]. Among other results, they showed that every connected $3-\gamma$-critical graph of even order contains a perfect matching. Wojcicka [27] subsequently studied hamiltonian properties of domination critical graphs and showed every connected $3-\gamma$-critical graph on at least seven vertices is traceable. Favaron et al. [9], Flandrin et al. [10] and Tian et al. [26] proved further that all connected $3-\gamma$-critical graphs with minimum degree at least 2 are hamiltonian. Motivated in part by these results, Sumner and Wojcicka (Chapter 16 in [12]) conjectured in 1998 that all $(k-1)$-connected k-γ-critical graphs are hamiltonian for all k ≥ 4. However, their conjecture was disproved seven years later by Yuansheng et al. [29] who constructed a 3-connected $4-\gamma$-critical non-hamiltonian graph containing 13 vertices. On the positive side, Kaemawichanurat and Caccetta [19] proved the Sumner-Wojcicka Conjecture is true if $k=4$ and the graphs are claw-free.

Kaemawichanurat [15] initiated a study of connected domination critical graphs. Hamiltonian properties of connected domination critical graphs were subsequently studied by Kaemawichanurat, Caccetta and Ananchuen [20] who showed that every 2-connected k-γ_c-critical graph is hamiltonian for all k ∈ [3]. Further, they constructed k-γ_c-critical graphs that are non-hamiltonian for all k ≥ 4. Recently, Kaemawichanurat and Caccetta [19] proved that every 2-connected $4-{\gamma }_c$-critical claw-free graph is hamiltonian, and they constructed 2-connected k-γ_c-critical claw-free graphs that are non-hamiltonian for all k ≥ 5. For 5 ≤ k ≤ 6, they proved that every 3-connected k-γ_c-critical claw-free graph is hamiltonian. Recall that ζ(G) denotes the number of cut-vertices of G, and that if the graph G is clear from the context, we simply write ζ instead of ζ(G). Kaemawichanurat and Ananchuen [18] showed that a connected domination critical graph cannot have too many cut-vertices.

Theorem 1

([18]) For k ≥ 2, every k-γ_c-critical graph has at most $k-2$ cut-vertices, that is, $\zeta \le k-2$.