Path factors and parallel knock-out schemes of almost claw-free graphs

An H₁,{H₂}-factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H₁ and all other components (if there are any) isomorphic to the graph H₂. We completely characterise the class of connected almost claw-free graphs that have a P₇,{P₂}-factor, where P₇ and P₂ denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation, we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost claw-free graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n^5.376) time algorithm for claw-free graphs on n vertices).     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

Bounds on total domination in claw-free cubic graphs

A set S of vertices in a graph G is a total dominating set, denoted by TDS, of G if every vertex of G is adjacent to some vertex in S (other than itself). The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). If G does not contain K_1,3 as an induced subgraph, then G is said to be claw-free. It is shown in [D. Archdeacon, J. Ellis-Monaghan, D. Fischer, D. Froncek, P.C.B. Lam, S. Seager, B. Wei, R. Yuster, Some remarks on domination, J. Graph Theory 46 (2004) 207-210.] that if G is a graph of order n with minimum degree at least three, then γ_t(G)?n/2. Two infinite families of connected cubic graphs with total domination number one-half their orders are constructed in [O. Favaron, M.A. Henning, C.M. Mynhardt, J. Puech, Total domination in graphs with minimum degree three, J. Graph Theory 34(1) (2000) 9-19.] which shows that this bound of n/2 is sharp. However, every graph in these two families, except for K₄ and a cubic graph of order eight, contains a claw. It is therefore a natural question to ask whether this upper bound of n/2 can be improved if we restrict G to be a connected cubic claw-free graph of order at least 10. In this paper, we answer this question in the affirmative. We prove that if G is a connected claw-free cubic graph of order n?10, then γ_t(G)?5n/11.     

Connected even factors in claw-free graphs

A connected even [2,2s]-factor of a graph G is a connected factor with all vertices of degree i (i=2,4,…,2s), where s?1 is an integer. In this paper, we show that every supereulerian K_1,s-free graph (s?2) contains a connected even [2,2s-2]-factor, hereby generalizing the result that every 4-connected claw-free graph has a connected [2,4]-factor by Broersma, Kriesell and Ryjacek.     

On Hamiltonicity of 3-Connected Claw-Free Graphs

Lai, Shao and Zhan (J Graph Theory 48:142–146, 2005) showed that every 3-connected N ₂-locally connected claw-free graph is Hamiltonian. In this paper, we generalize this result and show that every 3-connected claw-free graph G such that every locally disconnected vertex lies on some induced cycle of length at least 4 with at most 4 edges contained in some triangle of G is Hamiltonian. It is best possible in some sense.     

Subdivisions of K5 in Graphs Embedded on Surfaces With Face‐Width at Least 5

We prove that if G is a 5‐connected graph embedded on a surface Σ (other than the sphere) with face‐width at least 5, then G contains a subdivision of K₅. This is a special case of a conjecture of P. Seymour, that every 5‐connected nonplanar graph contains a subdivision of K₅. Moreover, we prove that if G is 6‐connected and embedded with face‐width at least 5, then for every v ∈ V(G), G contains a subdivision of K₅ whose branch vertices are v and four neighbors of v.     

Prism‐hamiltonicity of triangulations

The prism over a graph G is the Cartesian product G □ K₂ of G with the complete graph K₂. If the prism over G is hamiltonian, we say that G is prism‐hamiltonian. We prove that triangulations of the plane, projective plane, torus, and Klein bottle are prism‐hamiltonian. We additionally show that every 4‐connected triangulation of a surface with sufficiently large representativity is prism‐hamiltonian, and that every 3‐connected planar bipartite graph is prism‐hamiltonian. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 181–197, 2008     

Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t(G). A graph is claw-free if it does not contain K_1,3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G∉{C₃,C₅, C₆, C₁₀}, then γ_t(G)?4n/7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t(G)?(n+2)/2 and we characterize those graphs for which γ_t(G)=⌊(n+2)/2⌋.     

Forbidden induced subgraphs for star-free graphs

Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In Aldred et al. (2010) [1], it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least 2 and maximum degree at least 3. In the same work, it was also asked whether there are other families with the same property. In this paper, we answer this question by solving a wider problem. We consider not only claw-free graphs but the more general class of star-free graphs. Concretely, given t≥3, we characterize all the graph families H such that every large enough H-free connected graph is K_1,t-free. Additionally, for the case t=3, we show the families that one gets when adding the condition ∣H∣≤k for each positive integer k.     

Paired-Domination in Claw-Free Cubic Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by _pr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K_1,3 or K₄–e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K_1,3,K₄–e,C₄)-free, then _pr(G)3n/8; (ii) if G is claw-free and diamond-free, then _pr(G)2n/5; (iii) if G is claw-free, then _pr(G)n/2. In all three cases, the extremal graphs are characterized.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. This paper was written while the second author was visiting the Laboratoire de Recherche en Informatique (LRI) at the Université de Paris-Sud in July 2002. The second author thanks the LRI for their warm hospitality     

Hamiltonian connected hourglass free line graphs

Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K₅-E(C₄), where C₄ is a cycle of length 4 in K₅. In Broersma et al. [On factors of 4-connected claw-free graphs, J. Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph, is hamiltonian connected.     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Hamiltonian Connectedness in Claw-Free Graphs

If every vertex cut of a graph G contains a locally 2-connected vertex, then G is quasilocally 2-connected. In this paper, we prove that every connected quasilocally 2-connected claw-free graph is Hamilton-connected.     

Non-traceability of large connected claw-free graphs

Let G be a connected claw-free graph on n vertices. Let ς₃(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ς₃(G) ≥ n − 3 then G is traceable or else G is one of graphs G_n each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K₃. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer ν(k) such that if ς₃(G) ≥ n − k, n > ν(k) and G is non-traceable, then G is a factor of a graph G_n. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G = (V, E) (which is known to be NP-complete) has linear time complexity O(|E|) provided that ς₃(G) ≥ . This contrasts sharply with known results on NP-completeness among dense graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 75–86, 1998     

Randomly matchable graphs

A graph is defined to be randomly matchable if every matching of G can be extended to a perfect matching. It is shown that the connected randomly matchable graphs are precisely K_2n and K_n,n (n ≥ 1).     

Hamilton weights and Petersen minors*

A (1, 2)‐eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3‐connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K₄ by a series of Δ?Y‐operations. As a byproduct of the proof of the main theorem, we also prove that if G is a permutation graph and w is a (1,2)‐eulerian weight of G such that (G, w) is a critical contra pair, then the Petersen minor appears “almost everywhere” in the graph G. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 197–219, 2001     

Tank-Ring Factors in Supereulerian Claw-Free Graphs

A graph G has a tank-ring factor F if F is a connected spanning subgraph with all vertices of degree 2 or 4 that consists of one cycle C and disjoint triangles attaching to exactly one vertex of C such that every component of G ? C contains exactly two vertices. In this paper, we show the following results. (1) Every supereulerian claw-free graph G with 1-hourglass property contains a tank-ring factor. (2) Every supereulerian claw-free graph with 2-hourglass property is Hamiltonian.     

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

On a Closure Concept in Claw-Free Graphs

IfGis a claw-free graph, then there is a graphcl(G) such that (i) Gis a spanning subgraph ofcl(G), (ii) cl(G) is a line graph of a triangle-free graph, and (iii) the length of a longest cycle inGand incl(G) is the same. A sufficient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected claw-free graphs are obtained as corollaries.     

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U V(G) ,let N(U)=Uu,∈UN(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K1.3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al. : Let G be a 2-connected claw-free graph of order n,and d(u) d(v) d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that, for any three positive integers s ,t and w,such that if G is a (s t w-1)connected claw-free graph of order n,and d(S) d(T) d(W)＞n-(s t w) for every three disjoint independent vertex sets S,T,W with |S |=s, |T|=t, |W|=w,and S∪T∪W is also independent ,then G is Hamiltonian. Other related results are obtained too.