Marisa Gutierrez

The complexity of clique graph recognition

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C(G) the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of C(G). Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.

Clique graph recognition is NP-complete

A complete set of a graph G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. Denote by the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of . Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. We prove that the clique graph recognition problem is NP-complete.

Cliques and extended triangles. A necessary condition for planar clique graphs

By generalizing the idea of extended triangle of a graph, we succeed in obtaining a common framework for the result of Roberts and Spencer about clique graphs and the one of Szwarcfiter about Helly graphs. We characterize Helly and 3-Helly planar graphs using extended triangles. We prove that if a planar graph G is a clique graph, then every extended triangle of G must be a clique graph. Finally, we show the extended triangles of a planar graph which are clique graphs. Any one of the obtained characterizations are tested in O(n2) time.

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.

On the Clique Operator

The clique operator K maps a graph G into its clique graph, which is the intersection graph of the (maximal) cliques of G. Among all the better studied graph operators, K seems to be the richest one and many questions regarding it remain open. In particular, it is not known whether recognizing a clique graph is in P. In this note we describe our progress toward answering this question. We obtain a necessary condition for a graph to be in the image of K in terms of the presence of certain subgraphs A and B. We show that being a clique graph is not a property that is maintained by addition of twins. We present a result involving distances that reduces the recognition problem to graphs of diameter at most two. We also give a constructive characterization of K−1(G) for a fixed but generic G.

Pebbling in Split Graphs

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of 2 pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles

Recognizing vertex intersection graphs of paths on bounded degree trees

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On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

so that, from any initial configuration of

Pebbling in 2-Paths

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On the correspondence between tree representations of chordal and dually chordal graphs

pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding whether a given configuration on a particular graph can reach a specified target is \sf NP-complete, even for diameter