Claw-free Graphs Research Articles

Spectral conditions for traceability of connected claw-free graphs

In this paper, we establish sufficient conditions for a connected claw-free graph to be traceable in terms of the signless Laplacian spectral radius of the graph or Laplacian eigenvalues of its complements, respectively.

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k−s) and d(x)+d(y) ≥ n−2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K3s and k − s disjoint K4s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.

Maximum weight independent set for [formula omitted]claw-free graphs in polynomial time

For a finite, simple, undirected graph G with a vertex weight function, the Maximum Weight Independent Set (MWIS) problem asks for an independent vertex set of G with maximum weight. MWIS is a well-known NP-hard problem of fundamental importance. For claw-free graphs, MWIS can be solved in polynomial time—in 1980, the first two such algorithms were independently found by Minty for MWIS and by Sbihi for the unweighted case. For a constant ℓ≥2, let ℓG denote the disjoint union of ℓ copies of G.In this paper, using a dynamic programming approach (inspired by Farber’s result about MWIS for 2K2-free graphs), we show that for any fixed ℓ, MWIS can be solved in polynomial time for ℓclaw-free graphs. This solves the open cases for MWIS on (P3+claw)-free graphs and on (2K2+claw)-free graphs and extends known results for claw-free graphs, ℓK2-free graphs, (K2+claw)-free graphs, and ℓP3-free graphs.

Hamiltonian properties of 3-connected {claw,hourglass}-free graphs

We show that some sufficient conditions for hamiltonian properties of claw-free graphs can be substantially strengthened under an additional assumption that G is hourglass-free (where hourglass is the graph with degree sequence 4,2,2,2,2).Let G be a 3-connected claw-free and hourglass-free graph of order n. We show that (i)if G is P20-free, Z18-free, or N2i,2j,2k-free with i+j+k≤9, then G is hamiltonian,(ii)if G is P12-free, then G is Hamilton-connected,(iii)G contains a cycle of length at least min{σ12(G),n}, unless L−1(cl(G)) has a nontrivial contraction to the Petersen graph,(iv)if σ13(G)≥n+1, then G is hamiltonian, unless L−1(cl(G)) has a nontrivial contraction to the Petersen graph.Here Pi denotes the path on i vertices, Zi (Ni,j,k) denotes the graph obtained by attaching a path of length i≥1 (three vertex-disjoint paths of lengths i,j,k≥1) to a triangle, σk(G) denotes the minimum degree sum over all independent sets of size k, and L−1(cl(G)) is the line graph preimage of the closure of G.

Depth first search in claw-free graphs

Optimization problems concerning the vertex degrees of spanning trees of connected graphs play an extremely important role in network design. Minimizing the number of leaves of the spanning trees is NP-hard, since it is a generalization of the problem of finding a hamiltonian path of the graph. Moreover, Lu and Ravi (The power of local optimization: approximation algorithms for maximum-leaf spanning tree (DRAFT), CS-96-05, Department of Computer Science, Brown University, Providence, 1996) showed that this problem does not even have a constant factor approximation, unless \(\hbox {P}=\hbox {NP}\), thus properties that guarantee the existence of a spanning tree with a small number of leaves are of special importance. In this paper we are dealing with finding spanning trees with few leaves in claw-free graphs. We prove that all claw-free graphs have a DFS-tree, such that the leaves different from the root have no common neighbour, generalizing a theorem of Kano et al. (Ars Combin 103:137–154, 2012). The result also implies a strengthening of a result of Ainouche et al. (Ars Combin 29C:110–121, 1990).

The length of dominating cycle of claw-free graph

The length of dominating cycles is usually discussed in control problems. A dominating cycle of a graph G is a cycle C of G such that V(G)−V(C) is an independent set. In this article, we prove that for any claw-free graph G with δ(G)≥2, the length of longest dominating cycle is at least min{n,2|NC2(G)|−1}, where NC2(G) denotes the vertex set N(u)∪N(v) containing the minimum number of vertices for all vertices u,v with d(u,v)=2 in G.

Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset \(E'\) of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of \(E'\). We say that \(E'\) is a perfect edge dominating set of G, if every edge not in \(E'\) is dominated by exactly one edge of \(E'\). The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most \(d \ge 3\) and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a \(P_5\)-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a \(P_5\).

Properties of minimally [formula omitted]-tough graphs

A graph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases the toughness. Kriesell conjectured that for every minimally 1-tough graph the minimum degree δ(G)=2. We show that in every minimally 1-tough graph δ(G)≤n3+1. We also prove that every minimally 1-tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number t any graph can be embedded as an induced subgraph into a minimally t-tough graph.

Forbidden set of induced subgraphs for [formula omitted]-connected supereulerian graphs

Let H={H1,…,Hk} be a set of connected graphs. A graph is said to be H-free if it does not contain any member of H as an induced subgraph. We show that if the following statements hold, •|H|≥2,•K1,3∈H,•for any integer n0, every 2-connected H-free graph G of order at least n0 is supereulerian, i.e.,G has a spanning closed trail,then H∖{K1,3} contains an Ni,j,k or a path where Ni,j,k denotes the graph obtained by attaching three vertex-disjoint paths of lengths i,j,k≥0 to a triangle.As an application, we characterize all the forbidden triples H with K1,3∈H such that every 2-connected H-free graph is supereulerian. As a byproduct, we also characterize minimal 2-connected non-supereulerian claw-free graphs.

Forbidden subgraphs for chorded pancyclicity

We call a graph Gpancyclic if it contains at least one cycle of every possible length m, for 3≤m≤|V(G)|. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length 4,5,…,|V(G)|. In particular, certain paths and triangles with pendant paths are forbidden.

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H, let σt(H)=min{Σi=1tdH(vi)|{v1,v2,…,vt}is an independent set in H} and let Ut(H)=min{|⋃i=1tNH(vi)||{v1,v2,⋯,vt}is an independent set in H}. We show that for a given number ϵ and given integers p≥t>0, k∈{2,3} and N=N(p,ϵ), if H is a k-connected claw-free graph of order n>N with δ(H)≥3 and its Ryjác̆ek’s closure cl(H)=L(G), and if dt(H)≥t(n+ϵ)∕p where dt(H)∈{σt(H),Ut(H)}, then either H is Hamiltonian or G, the preimage of L(G), can be contracted to a k-edge-connected K3-free graph of order at most max{4p−5,2p+1} and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large:(i) If k=2, δ(H)≥3, and for a given t (1≤t≤4) dt(H)≥tn4, then either H is Hamiltonian or cl(H)=L(G) where G is a graph obtained from K2,3 by replacing each of the degree 2 vertices by a K1,s (s≥1). When t=4 and dt(H)=σ4(H), this proves a conjecture in Frydrych (2001).(ii) If k=3, δ(H)≥24, and for a given t (1≤t≤10) dt(H)>t(n+5)10, then H is Hamiltonian. These bounds on dt(H) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σt and Ut for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max{4p−5,2p+1} are fixed for given p, improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer.

Colouring squares of claw-free graphs

Is there some absolute ε>0 such that for any claw-free graph G, the chromatic number of the square of G satisfies χ(G2)≤(2−ε)ω(G)2, where ω(G) is the clique number of G? Erdős and Nešetřil asked this question for the specific case of G the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil.

On some domination colorings of graphs

In this paper, we are interested in four proper vertex colorings of graphs, with additional domination property. In the dominator colorings, strong colorings and strict strong colorings of a graph G, every vertex has to dominate at least one color class. Conversely, in the dominated colorings of G, every color class has to be dominated by at least one vertex. We study arbitrary graphs as well as P4-sparse graphs, P5-free graphs, bounded treewidth graphs and claw-free graphs.

Vizing’s conjecture: A two-thirds bound for claw-free graphs

We show that for any claw-free graph G and any graph H, γ(G□H)≥23γ(G)γ(H), where γ(G) is the domination number of G.

Circumferences of 3-connected claw-free graphs, II

For a graph H, the circumference of H, denoted by c(H), is the length of a longest cycle in H. It is proved in Chen (2016) that if H is a 3-connected claw-free graph of order n with δ≥8, then c(H)≥min{9δ−3,n}. In Li (2006), Li conjectured that every 3-connected k-regular claw-free graph H of order n has c(H)≥min{10k−4,n}. Later, Li posed an open problem in Li (2008): how long is the best possible circumference for a 3-connected regular claw-free graph? In this paper, we study the circumference of 3-connected claw-free graphs without the restriction on regularity and provide a solution to the conjecture and the open problem above. We determine five families Fi (1≤i≤5) of 3-connected claw-free graphs which are characterized by graphs contractible to the Petersen graph and show that if H is a 3-connected claw-free graph of order n with δ≥16, then one of the following holds:(a) either c(H)≥min{10δ−3,n} or H∈F1.(b) either c(H)≥min{11δ−7,n} or H∈F1∪F2.(c) either c(H)≥min{11δ−3,n} or H∈F1∪F2∪F3.(d) either c(H)≥min{12δ−10,n} or H∈F1∪F2∪F3∪F4.(e) if δ≥23 then either c(H)≥min{12δ−7,n} or H∈F1∪F2∪F3∪F4∪F5.This is also an improvement of the prior results in Chen (2016), Lai et al. (2016), Li et al. (2009) and Mathews and Sumner (1985).

Toughness and hamiltonicity in almost claw-free graphs

Toughness and hamiltonicity in almost claw-free graphs

Double-critical graph conjecture for claw-free graphs

A connected graph G with chromatic number t is double-critical if G∖{x,y} is (t−2)-colorable for each edge xy∈E(G). The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph G with chromatic number t is double-critical, then G is the complete graph on t vertices. This has been verified for t≤5, but remains open for t≥6. In this paper, we first prove that if G is a non-complete, double-critical graph with chromatic number t≥6, then no vertex of degree t+1 is adjacent to a vertex of degree t+1, t+2, or t+3 in G. We then use this result to show that the Double-Critical Graph Conjecture is true for double-critical graphs G with chromatic number t≤8 if G is claw-free.

Computing large independent sets in a single round

Independent sets play a central role in distributed algorithmics. We examine here the minimal requirements for computing non-trivial independent sets. In particular, we focus on algorithms that operate in a single communication round. A classic result of Linial shows that a constant number of rounds does not suffice to compute a maximal independent set. We are therefore interested in the size of the solution that can be computed, especially in comparison to the optimal. Our main result is a randomized one-round algorithm that achieves poly-logarithmic approximation on graphs of polynomially bounded-independence. Specifically, we show that the algorithm achieves the Caro-Wei bound (an extension of the Turán bound for independent sets) in general graphs up to a constant factor, and that the Caro-Wei bound yields a poly-logarithmic approximation on bounded-independence graphs. The algorithm uses only a single bit message and operates in a beeping model, where a node receives only the disjunction of the bits transmitted by its neighbors. We give limitation results that show that these are the minimal requirements for obtaining non-trivial solutions. In particular, a sublinear approximation cannot be obtained in a single round on general graphs, nor when nodes cannot both transmit and receive messages. We also show that our analysis of the Caro-Wei bound on polynomially bounded-independence graphs is tight, and that the poly-logarithmic approximation factor does not extend to mathrm {O}(1)-claw free graphs.

On line graphs of subcubic triangle-free graphs

Line graphs constitute a rich and well-studied class of graphs. In this paper, we focus on three different topics related to line graphs of subcubic triangle-free graphs. First, we show that any such graph G has an independent set of size at least 3|V(G)|∕10, the bound being sharp. As an immediate consequence, we have that any subcubic triangle-free graph G, with ni vertices of degree i, has a matching of size at least 3n1∕20+3n2∕10+9n3∕20. Then we provide several approximate min-max theorems relating cycle-transversals and cycle-packings of line graphs of subcubic triangle-free graphs. This enables us to prove Jones’ Conjecture for claw-free graphs with maximum degree 4. Finally, we concentrate on the computational complexity of Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path for subclasses of line graphs of subcubic triangle-free graphs.

Maximum number of edges in claw-free graphs whose maximum degree and matching number are bounded

We determine the maximum number of edges that a claw-free graph can have, when its maximum degree and matching number are bounded. This is a famous problem that has been studied on general graphs, and for which there is a tight bound. The graphs achieving this bound contain in most cases an induced copy of K1,3, the claw, which motivates studying the question on claw-free graphs. Note that on general graphs, if one of the mentioned parameters is not bounded, then there is no upper bound on the number of edges. We show that on claw-free graphs, bounding the matching number is sufficient for obtaining an upper bound on the number of edges. The same is not true for the degree, as a long path is claw-free. We give exact tight formulas for both when only the matching number is bounded and when both parameters are bounded. We also construct claw-free graphs whose edge numbers match the given bounds.