Claw-free Graphs Research Articles

Hamiltonicity and restricted degree conditions on induced subgraphs in claw-free graphs

For a graph S, a vertex with degree one in S is an end-vertex of S and an edge is a pendant edge if one of its ends is an end-vertex. A net N is a graph obtained from a K3 by adding a pendant edge at each vertex of the K3. A P6 is a path on 6 vertices. In this paper, we solve two conjectures and prove that for 2-connected claw-free graphs H and for a fixed graph S∈{N,P6}, if the degree at each end-vertex of every induced copy of S is at least (|V(H)|−2)∕3, then H is Hamiltonian. The case for S=N was conjectured by Broersma (1993); the case for S=P6 was conjectured by C̃ada et al. (2016).

Hamiltonian claw-free graphs involving induced cycles

The hamiltonian problem is an important topic in structural graph theory, which is closely related to Four Color Problem. Hence lots of graph scholars are dedicated to this topic. There are many authors working for finding some sufficient conditions for hamiltonian property of graphs. Let G be a claw-free graph with n vertices and δ(G)≥3. In this paper, we show that if G has an induced cycle of length more than (4n – 2δ(G)–4)(δ(G)+2)−1, then G is hamiltonian. The result is best possible if δG is 3 or 4.

Edge-Distinguishing of Star-Free Graphs

The distinguishing index $D'(G)$ of a graph $G$ is the least number $k$ such that $G$ has an edge colouring with $k$ colours that is only preserved by the trivial automorphism. Pilśniak proved that a connected, claw-free graph has the distingushing index at most three. In this paper, we show that the distingushing index of a connected, claw-free graph with at least six vertices is bounded from above by two. We also consider more general graphs in this sense. Namely, we prove that if $G$ is a connected, $K_{1,s}$-free graph of order at least six, then $D'(G) \leq s-1$.

Z-domination game

The Z-domination game is a variant of the domination game in which each newly selected vertex u in the game must have a not yet dominated neighbor, but after the move all vertices from the closed neighborhood of u are declared to be dominated. The Z-domination game is the fastest among the five natural domination games. The corresponding game Z-domination number of a graph G is denoted by γZg(G). It is proved that the game domination number and the game total domination number of a graph can be expressed as the game Z-domination number of appropriate lexicographic products. Graphs with a Z-insensitive property are introduced and it is proved that if G is Z-insensitive, then γZg(G) is equal to the game domination number of G. Weakly claw-free graphs are defined and proved to be Z-insensitive. As a consequence, γZg(Pn) is determined, thus sharpening an earlier related approximate result. It is proved that if γZg(G) is an even number, then γZg(G) is strictly smaller than the game L-domination number. On the other hand, families of graphs are constructed for which all five game domination numbers coincide. Graphs G with γZg(G)=γ(G) are also considered and computational results which compare the studied invariants in the class of trees on at most 16 vertices reported.

Domination versus edge domination

We propose the conjecture that the domination number γ(G) of a Δ-regular graph G with Δ≥1 is always at most its edge domination number γe(G), which coincides with the domination number of its line graph. We prove that γ(G)≤1+2(Δ−1)Δ2Δγe(G) for general Δ≥1, and γ(G)≤76−1204γe(G) for Δ=3. Furthermore, we verify our conjecture for cubic claw-free graphs.

Partitioning Claw-Free Subcubic Graphs into Two Dominating Sets

A dominating set in a graph G is a set $$S\subseteq V(G)$$ such that every vertex in $$V(G){\setminus } S$$ has at least one neighbor in S. Let G be an arbitrary claw-free graph containing only vertices of degree two or three. In this paper, we prove that the vertex set of G can be partitioned into two dominating sets $$V_1$$ and $$V_2$$ such that for $$i=1,2$$, the subgraph of G induced by $$V_i$$ is triangle-free and every vertex $$v\in V_i$$ also has at least one neighbor in $$V_i$$ if v has degree three in G. This gives an affirmative answer to a problem of Bacso et al. and generalizes a result of Desormeaux et al.

Disconnected cuts in claw-free graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. This problem is known to be NP-hard on general graphs. We prove that it is polynomial-time solvable on claw-free graphs, answering a question of Ito et al. (TCS 2011). The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Mind the independence gap

The independence gap of a graph was introduced by Ekim et al. in 2018 as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set.We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most k is polynomial-time solvable if k is fixed and co-NP-complete if k is part of input. We also investigate the complexity of the independent set and independent domination problems in graph classes related to independence gap. In particular, we show that the two problems are NP-complete in the class of graphs of independence gap at most one and that the weighted independent set problem is polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomial time in the class of {claw, 2P3}-free graphs.

Separation routine and extended formulations for the stable set problem in claw-free graphs

The maximum weighted stable set problem in claw-free graphs is a well-known generalization of the maximum weighted matching problem, and a classical problem in combinatorial optimization. In spite of the recent development of fast(er) combinatorial algorithms and some progresses in the characterization of the corresponding stable set polytope, the problem of “providing a decent linear description” for this polytope (Grotschel et al. in Geometric algorithms and combinatorial optimization, Springer, New York, 1988) is still open. The main contribution of this paper is to propose an algorithmic answer to that question by providing a polynomial-time and computationally attractive separation routine for the stable set polytope of claw-free graphs, that only requires a combinatorial decomposition algorithm, the solution of (moderate sized) compact linear programs, and Padberg and Rao’s algorithm for separating over the matching polytope. In particular, it is a generalization of the latter and avoids the heavy computational burden of resorting to the ellipsoid method, on which the only poly-time separation routine known so far relied. Besides, our separation routine comes with a ‘small’ (but not polynomial) extended linear programming formulation and a procedure to derive a linear description of the stable set polytope of claw-free graphs in the original space.

Power domination in regular claw-free graphs

In this paper, we show that the power domination number of a connected 4-regular claw-free graph on n vertices is at most n+15, and the bound is sharp. The statement disproves the conjecture presented by Dorbec et al. (2013).

A [formula omitted]-approximation of Vizing’s conjecture for claw-free graphs

Vizing’s conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. We prove that for any claw-free graph G and an arbitrary graph H, the inequality γ(G□H)≥34γ(G)γ(H) always holds.

Clique coverings and claw-free graphs

Let C be a clique covering for E(G) and let v be a vertex of G. The valency of vertex v (with respect to C), denoted by valC(v), is the number of cliques in C containing v. The local clique cover number of G, denoted by lcc(G), is defined as the smallest integer k, for which there exists a clique covering for E(G) such that valC(v) is at most k, for every vertex v∈V(G). In this paper, among other results, we prove that if G is a claw-free graph, then lcc(G)+χ(G)≤n+1.

Strong chromatic index of [formula omitted]-free graphs

A strong edge-coloring of a graph G is a coloring of the edges of G such that each color class is an induced matching. The strong chromatic index of G is the minimum number of colors in a strong edge-coloring of G.We show that the strong chromatic index of a claw-free graph with maximum degree Δ is at most 1.125Δ2+Δ, which confirms the conjecture of Erdős and Nešetřil from 1985 for this class of graphs for Δ≥12.We also prove an upper bound of 2−1t−2Δ2 on strong chromatic index of K1,t-free graphs with maximum degree Δ for all t≥4 and give an improved result 1.625Δ2 for unit disk graphs.

On sufficient degree conditions for traceability of claw-free graphs

We present new results on the traceability of claw-free graphs. In particular, we consider sufficient minimum degree and degree sum conditions that imply that these graphs admit a Hamilton path, unless they have a small order or they belong to well-defined classes of exceptional graphs. Our main result implies that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥3 is traceable if the degree sum of any set of t independent vertices of G is at least t(n+6)6, where t∈{1,2,…,6}, and that this lower bound n+66 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with minimum degree δ(G)≥22 is traceable if the degree sum of any set of t independent vertices of G is at least t(2n−5)14, where t∈{1,2,…,7}, unless G is a member of well-defined classes of exceptional graphs depending on t, and that this lower bound 2n−514 on the degree sums is asymptotically sharp. Our results also imply that a 2-connected claw-free graph G of sufficiently large order n with δ(G)≥18 is traceable if the degree sum of any set of 6 independent vertices is larger than n−6, and that this lower bound on the degree sums is sharp.

Cycle Traversability for Claw-Free Graphs and Polyhedral Maps

Let G be a graph, and $$v \in V(G)$$ and $$S \subseteq V(G)\setminus{v}$$ of size at least k. An important result on graph connectivity due to Perfect states that, if v and S are k-linked, then a (k−1)-link between a vertex v and S can be extended to a k-link between v and S such that the endvertices of the (k−1)-link are also the endvertices of the k-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets S1 and S2 are k-linked, then a t-link (t<k) between two disjoint sets S1 and S2 can be extended to a k-link between S1 and S2 such that the endvertices of the t-link are preserved in the k-link. Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices, but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that a 3-connected claw-free graph has a topological wheel minor Wk with k ≤ 5 if and only if it has a vertex of degree at least k. Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices, but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in surfaces having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.

A Note on Singular Edges and Hamiltonicity in Claw-Free Graphs with Locally Disconnected Vertices

An edge e of a graph G is called singular if it is not on a triangle; otherwise, e is nonsingular. A vertex is called singular if it is adjacent to a singular edge; otherwise, it is called nonsingular. We prove the following. Let G be a connected claw-free graph such that every locally disconnected vertex x ∈ V(G) satisfies the following conditions: Then G is either hamiltonian, or G is the line graph of the graph obtained from $$K_{2,3}$$ by attaching a pendant edge to its each vertex of degree two. Some results on forbidden subgraph conditions for hamiltonicity in 3-connected claw-free graphs are also obtained as immediate corollaries

An $${\mathcal {O}}(n^2 \log {n})$$ algorithm for the weighted stable set problem in claw-free graphs

A graph G(V, E) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into {claw, net}-free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in $$\mathcal{O}(|V|(|V| \log |V| + |E|))$$ time. In this paper, we describe a direct decomposition of a claw-free graph into {claw, net}-free strips and strips with stability number at most three which can be performed in $$\mathcal{O}(|V|^2)$$ time. In two companion papers we showed that the MWSS problem can be solved in $$\mathcal{O}(|E| \log |V|)$$ time in claw-free graphs with $$\alpha (G) \le 3$$ and in $$\mathcal{O}(|V| \sqrt{|E|})$$ time in {claw, net}-free graphs with $$\alpha (G) \ge 4$$ . These results prove that the MWSS problem in a claw-free graph can be solved in $$\mathcal{O}(|V|^2 \log |V|)$$ time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs.

On a Vizing-type Integer Domination Conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $\gamma_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Bre\v{s}ar, Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{\{k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{2\}$-domination number, $\gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{\{2\}}(G\square H)\ge \gamma_{R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

2-connected Hamiltonian claw-free graphs involving degree sum of adjacent vertices

Spanning Trails in a 2-Connected Graph

In this article we prove the following: Let $G$ be a $2$-connected graph with circumference $c(G)$. If $c(G)\leq 5$, then $G$ has a spanning trail starting from any vertex, if $c(G)\leq 7$, then $G$ has a spanning trail.&#x0D; As applications of this result, we obtain the following.&#x0D; &#x0D; &#x0D; Every $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex with the exception of six graphs.&#x0D; &#x0D; &#x0D; Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\emptyset$ and $|S|\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $|S|\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$.&#x0D; &#x0D; Every $2$-connected claw-free graph $G$ with order $n$ and minimum degree $\delta(G)&gt; \frac{n}{7}+4\geq 23$ is traceable or belongs to two exceptional families of well-defined graphs, and moreover, if $\delta(G)&gt; \frac{n}{6}+4\geq 13$, then $G$ is traceable.&#x0D; &#x0D; All above results are sharp in a sense.