R.J. Faudree, E. Flandrin, and Z. Ryj a cek, Claw-free graphs---a survey, Discrete Math., 164 (1997), pp. 87--147.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....zeros is the claw K 1;3 = i i i P P P r r r r This suggests the conjecture: Conjecture 2.5 (Hamidoune [25] Stanley [62] If G is claw free (i.e. has no induced subgraph K 1;3 ) then all the zeros of Z(w) are negative real. For possibly useful background on claw free graphs, see [19]; see also [22] For interesting results along vaguely related lines, see [47, 48] Finally, one may ask: What happens when the fugacity w is not small In particular, one would like to understand the uniqueness or nonuniqueness of the infinitevolume Gibbs measure as a function of w, the nature ....

Faudree, R., Flandrin, E. and Ryj'acek, Z. (1997) Claw-free graphs --- a survey. Discrete Math. 164, 87--147.

....[52] We refer to the sources for additional terminology and details. 11 4 Hamiltonicity of claw free graphs During the last two decades many results on hamiltonian properties of claw free graphs (i.e. graphs that do not contain K 1,3 as an induced subgraph) have appeared. We refer the reader to [29] for a recent survey. Most of these results involve su#cient conditions in terms of degrees, neighborhoods, forbidden subgraphs or (local) connectivity. In this section we will discuss several recent developments on hamiltonicity of claw free graphs. We start with the earliest minimum degree ....

....can be relaxed if we add the condition that the graph under consideration is claw free (and 2 connected) Theorem 22 ( Matthews and Sumner [60] Every 2 connected claw free graph G with #(G) # 1 3 (n 2) is hamiltonian. Theorem 22 has been generalized in several directions. We refer to [29] for a survey, and come back with the most recent developments on minimum degree conditions later. 4.1 On two conjectures and a closure technique Most of the results in this section are motivated by the following two conjectures. Conjecture 23 ( Thomassen [67] Every 4 connected line graph is ....

R.J. Faudree, E. Flandrin, and Z. Ryjacek, Claw-free graphs - a survey. Discrete Math. 164 (1997) 87--147.

....delta matroid parity problem [10] In this paper, we will deal with the maximum weight stable set problem for claw free graphs. We call the complete bipartite graph K 1;3 a claw. A graph is said to be claw free if it does not contain a vertex induced subgraph which is isomorphic to a claw (see [4] for a survey on claw free graphs) The claw is one of the forbidden subgraphs of line graphs [1] That is, the line graphs are claw free, and the maximum weight stable set problem for claw free graphs is a generalization of the maximum weight matching problem. Up to date, three polynomial time ....

R. Faudree, E. Flandrin and Z. Ryj'acek, Claw-free graphs --- A survey, Discrete Math. 164 (1997), 87-147.

....checks if a graph has a diamond and produces one if it does. Proof. Take D = p e. 2 4 Recognition of claw free graphs The importance of claw free graphs follows from matching properties, line graphs, hamiltonian properties and the polynomial time algorithm for computing the independence number [4, 7]. However, although many characterizations are known, there is no fast recognition algorithm known. Even the extensive survey on clawfree graphs in [4] mentiones only a O(n 3:5 ) recognition algorithm. We present in this section an O(e ff 1 2 ) O(e 1:69 ) recognition algorithm. Definition ....

....follows from matching properties, line graphs, hamiltonian properties and the polynomial time algorithm for computing the independence number [4, 7] However, although many characterizations are known, there is no fast recognition algorithm known. Even the extensive survey on clawfree graphs in [4] mentiones only a O(n 3:5 ) recognition algorithm. We present in this section an O(e ff 1 2 ) O(e 1:69 ) recognition algorithm. Definition 3 A claw is a graph isomorphic to the graph depicted in Figure 1 on the right. A graph is claw free if it does not have an induced subgraph isomorphic ....

Faudree, R., E. Flandrin and Z. Ryja#cek, Claw-free graphs--A survey. Manuscript.

....Stronger bounds are given in the special case of claw free graphs (i.e. r = 3) Sharpness examples are also presented. c fl John Wiley Sons, Inc. 1. INTRODUCTION Claw free graphs have been a subject of interest of many authors in the last years (see e.g. a recent survey by R. Faudree et al. [6]) For this class of graphs we investigate problems which have their origin in the theory of planar graphs. Throughout the paper we use the most common graph theoretical terminology. For the concepts not defined here we refer to [1] A graph G is called K 1;r free if there is no induced subgraph ....

R.Faudree, E. Flandrin , Z.Ryj'acek, Claw-free graphs - a survey, Discrete Math. 169(1997) 87-147 14 JOURNAL OF GRAPH THEORY

....graph has many global consequences. In this paper, we follow up in this direction by showing that restricting ff i (G) only at a few distance levels implies a restriction on the global independence number ff(G) For more related results on claw free graphs we refer the reader to survey paper [2]. 2 Main results For any integers r; t 2 we set S r;t = fGj ff 1 (G) r; ff 2 (G) tg. Note that all classes S 2;t are subclasses of the class of claw free graphs, and S 2;2 is the family of distance claw free 2 graphs, introduced in [4] For any integers k; i, 0 i b k 2 c, B k;i denotes ....

Faudree, R.J.; Flandrin, E.; Ryj'acek, Z.: Claw-free graphs - a survey. Discrete Mathematics 164 (1997), 87-147.

....9 in Theorems 7 10 and in Corollary 11 can be checked in polynomial time. On the other hand, it is known that the decision whether G is hamiltonian is NP complete even in line graphs (see [2] or, for more information on complexity results in claw free graphs, Chapter 5 of the survey paper [8]) 4. In the proofs of Theorems 7 9, the fact that the classes considered are stable allows to assume that all graphs under consideration are closed (i.e. are line graphs of triangle free graphs) and to use the structural information given by this fact to reduce the number of situations to be ....

Faudree, R.J., Flandrin, E., Ryj'acek, Z.: Claw-free graphs - a survey. Discrete Mathematics 164 (1997), 87-147.

....Stronger bounds are given in the special case of claw free graphs (i.e. r = 3) Sharpness examples are also presented. c fl John Wiley Sons, Inc. 1. INTRODUCTION Claw free graphs have been a subject of interest of many authors in the last years (see e.g. a recent survey by Faudree et al. [6]) For this class of graphs we investigate problems which have their origin in the theory of planar graphs. Throughout the paper we use the most common graph theoretical terminology. For the concepts not defined here we refer to [1] A graph G is called K 1;r free if there is no induced subgraph ....

R.Faudree, E. Flandrin , Z.Ryj'acek, Claw-free graphs - a survey, Discrete Math. 169(1997) 87-147

....of v) A vertex v 2 V (G) is locally connected if hN(v)i is connected, and the graph G is locally connected if all vertices of G are locally connected. 2. INTRODUCTION During the last two decades many results on hamiltonian properties of claw free graphs have appeared. We refer the reader to [4] for a recent survey. Most of these results involve sufficient conditions in terms of degrees, neighborhoods, forbidden subgraphs or (local) connectivity. These conditions are significantly weaker than the corresponding sufficient conditions for general graphs. Nevertheless, these conditions are ....

....w, where w 2 N(u) N(v) 7) fu; vg is a 3 pair in G Gamma w, and there are at least two internally disjoint (u; v) paths of length 3 in G Gamma w, where w 2 N(u) N(v) Most of the corollaries that follow have been generalized or extended, sometimes in several directions. We refer to [4] for more information. The corollaries that are stated without proof are immediate. Corollary 1.1. 6] If ffi(G) 1 3 (n Gamma 2) then G is hamiltonian. 3 Corollary 1.2. 8] If jN(u) N(v)j 2 for all 2 pairs fu; vg V (G) then G is hamiltonian. Corollary 1.3. 7] If G is ....

Faudree, R.J.; Flandrin, E.; Ryj'acek, Z.: Claw-free graphs - a survey. Preprint, University of West Bohemia, 1994 (to appear).

....are not very dense. 4. 1 Definition and basic properties A graph G is claw free if G does not contain an induced subgraph isomorphic to the complete bipartite graph K 1;3 (referred to as a claw) For additional information on results on claw free graphs we refer the reader to the survey paper [33]. For a vertex x of a graph G the subgraph of G induced by the set of neighbors NG (x) of x is called the neighborhood of x (in G) We say that x is locally connected if its neighborhood is a connected graph. A locally connected vertex with a noncomplete neighborhood is called eligible and the ....

Faudree, R.J.; Flandrin, E.; Ryj'acek, Z.: Claw-free graphs - a survey. Discrete Math. 164 (1997), 87-147.

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R.J. Faudree, E. Flandrin, and Z. Ryj a cek, Claw-free graphs---a survey, Discrete Math., 164 (1997), pp. 87--147.

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R. J. Faudree, E. Flandrin and Z. Ryjacek, Claw-free graphs - a survey, Discrete Math. 164 (1997), 87-147.

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