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18
Clawfree Graphs. V. Global structure
, 2007
"... A graph is clawfree if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every clawfree graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this ..."
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A graph is clawfree if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every clawfree graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this “decomposition” theorem into a theorem describing the global structure of clawfree graphs.
Partial characterizations of cliqueperfect graphs II: diamondfree and Helly circulararc graphs
, 2005
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Decomposition of evenholefree graphs with star cutsets and 2joins
, 2008
"... In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e. chordless cycles of even length). These graphs are known as evenholefree graphs. We prove a decomposition theorem for evenholefree graphs, that uses star cutsets and 2joins. This is ..."
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In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e. chordless cycles of even length). These graphs are known as evenholefree graphs. We prove a decomposition theorem for evenholefree graphs, that uses star cutsets and 2joins. This is a significant strengthening of the only other previously known decomposition of evenholefree graphs, by Conforti, Cornuéjols, Kapoor and Vušković, that uses 2joins and star, double star and triple star cutsets. It is also analogous to the decomposition of Berge (i.e. perfect) graphs with skew cutsets, 2joins and their complements, by Chudnovsky, Robertson, Seymour and Thomas. The similarity between evenholefree graphs and Berge graphs is higher than the similarity between evenholefree graphs and simply oddholefree graphs, since excluding a 4hole, automatically excludes all antiholes of length at least 6. In a graph that does not contain a 4hole, a skew cutset reduces to a star cutset, and a 2join in the complement implies a star cutset, so in a way it was expected that evenholefree graphs can be decomposed with just the star cutsets and 2joins. A consequence of this decomposition theorem is a recognition algorithm for evenholefree graphs that is significantly faster than the previously known ones. Key words: Evenholefree graphs, star cutsets, 2joins, recognition algorithm, decomposition. 1
The robber strikes back
"... We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph G, writ ..."
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Cited by 5 (5 self)
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We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph G, written cc(G). We give bounds on cc(G) in terms of the cop number of G in the classes of bipartite graphs and diameter two, K1,mfree graphs. AMS 2010 Subject Classification: 05C57
Dominating set is fixed parameter tractable in clawfree graphs
 CoRR
"... We show that the DOMINATING SET problem parameterized by solution size is fixedparameter tractable (FPT) in graphs that do not contain the claw (K1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an a ..."
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We show that the DOMINATING SET problem parameterized by solution size is fixedparameter tractable (FPT) in graphs that do not contain the claw (K1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2O(k 2)nO(1) time and polynomial space to decide whether a clawfree graph on n vertices has a dominating set of size at most k. Note that this parameterization of DOMINATING SET is W [2]hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of DOMINATING SET is the class of Ki,jfree graphs, which exclude, for some fixed i, j ∈ N, the complete bipartite graph Ki,j as a subgraph. For i, j ≥ 2, the class of clawfree graphs and any class of Ki,jfree graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of DOMINATING SET is known to be fixedparameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the DOMINATING SET problem hard. More precisely, we show that for any t ≥ 4, the DOMINATING SET problem parameterized by the solution size is W [2]hard in graphs that exclude the tclaw K1,t as an induced subgraph. Our arguments also imply that the related CONNECTED DOMINATING SET and DOMINATING CLIQUE problems are W [2]hard in these graph classes. Finally, we show that for any t ∈ N, the CLIQUE problem parameterized by solution size, which is W [1]hard on general graphs, is FPT in tclawfree graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors. 1
2cliquebond of stable set polyhedra
, 2009
"... The 2bond is a generalization of the 2join where the subsets of nodes that are connected on each shore of the partition are not necessarily disjoint. If all the subsets are cliques we say that the 2bond is a 2cliquebond. We consider a graph G obtained as the 2cliquebond of two graphs G1 and G ..."
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The 2bond is a generalization of the 2join where the subsets of nodes that are connected on each shore of the partition are not necessarily disjoint. If all the subsets are cliques we say that the 2bond is a 2cliquebond. We consider a graph G obtained as the 2cliquebond of two graphs G1 and G2 and we study the polyhedral properties of the stable set polytope associated with this graph. In particular, we prove that a linear description of the stable set polytope of G is obtained by properly composing the linear inequalities describing the stable set polytopes of four graphs that are related to G1 and G2. We show how to apply the 2cliquebond composition to provide the complete linear description of large classes of graphs.
THE STABLE SET POLYTOPE OF CLAWFREE GRAPHS I: XXSTRIP COMPOSITION VERSUS GEAR COMPOSITION
, 2009
"... The gear composition builds a new graph G by substituting a suitable edge of a given graph H with a fixed graph named gear. Here we extend this definition to obtain an operation that is suitable to handle clawfree graphs. We call geared (fuzzy) line graphs the graphs obtained from (fuzzy) line grap ..."
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Cited by 1 (1 self)
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The gear composition builds a new graph G by substituting a suitable edge of a given graph H with a fixed graph named gear. Here we extend this definition to obtain an operation that is suitable to handle clawfree graphs. We call geared (fuzzy) line graphs the graphs obtained from (fuzzy) line graphs by repeated applications of the extended gear composition and we prove that these graphs form a significant subclass of clawfree graphs with stability number at least four. The proof is based on the decomposition theorem of Chudnovsky and Seymour [1, 2]. We also show how the extended gear composition generates facet defining inequalities for the stable set polytope of a geared graph G. In a sequel we prove that these facet defining inequalities yield the complete linear description of the stable set polytope of geared (fuzzy) line graphs.
LAZY COPS AND ROBBERS PLAYED ON GRAPHS
"... Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojak ..."
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Abstract. We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By investigating expansion properties, we provide asymptotically almost sure bounds on the lazy cop number of binomial random graphs G(n, p) for a wide range of p = p(n). By coupling the probabilistic method with a potential function argument, we also improve on the existing lower bounds for the lazy cop number of hypercubes. Finally, we provide an upper bound for the lazy cop number of graphs with genus g by using the GilbertHutchinsonTarjan separator theorem. 1.
Clawfree Graphs. III. Circular interval graphs
, 2003
"... Abstract Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points V from the circle; and make a graph with vertex set V in which two vertices are adjacent if they both belong to one of the interval ..."
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Abstract Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points V from the circle; and make a graph with vertex set V in which two vertices are adjacent if they both belong to one of the intervals. Such graphs are &quot;long circular interval graphs&quot;, and they form an important subclass of the class of all clawfree graphs. In this paper we characterize them by excluded induced subgraphs. This is a step towards the main goal of this series, to find a structural characterization of all clawfree graphs. This paper also gives an analysis of the clawfree graphs G with a clique the deletion of which disconnects G into two parts both with at least two vertices.