147 research outputs found
The icosahedron is clique divergent
AbstractA clique of a graph G is a maximal complete subgraph. The clique graph k(G) is the intersection graph of the set of all cliques of G. The iterated clique graphs are defined recursively by k0(G)=G and kn+1(G)=k(kn(G)). A graph G is said to be clique divergent (or k-divergent) if limn→∞|V(kn(G))|=∞. The problem of deciding whether the icosahedron is clique divergent or not was (implicitly) stated Neumann-Lara in 1981 and then cited by Neumann-Lara in 1991 and Larrión and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory
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.Facultad de Ciencias Exacta
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.Facultad de Ciencias Exacta
Beyond Helly graphs: the diameter problem on absolute retracts
Characterizing the graph classes such that, on -vertex -edge graphs in
the class, we can compute the diameter faster than in time is an
important research problem both in theory and in practice. We here make a new
step in this direction, for some metrically defined graph classes.
Specifically, a subgraph of a graph is called a retract of if it is
the image of some idempotent endomorphism of . Two necessary conditions for
being a retract of is to have is an isometric and isochromatic
subgraph of . We say that is an absolute retract of some graph class
if it is a retract of any of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary graph
classes. First, we show how to compute the diameter within absolute retracts of
bipartite graphs in randomized time. For the
special case of chordal bipartite graphs, it can be improved to linear time,
and the algorithm even computes all the eccentricities. Then, we generalize
these results to the absolute retracts of -chromatic graphs, for every fixed
. Finally, we study the diameter problem within the absolute retracts
of planar graphs and split graphs, respectively
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