Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) polygon C and a finite set P of points in general position in the complement of C, the visibility graph GC(P ) has a vertex for each point in P and an edge pq for any two points p and q in P that can see ...

Deciding whether a collection of unrooted trees is compatible is a fundamental problem in phylogenetics. Two different graph-theoretic characterizations of tree compatibility have recently been proposed. In one of these, tree compatibility is characterized in terms of the existence of a specific kind of triangulation in a structure known as the display graph. An alternative characterization exp...

The P4-sparse Graph Sandwich Problem asks, given two graphs G 1 = (V, E1) and G2 = (V, E2), whether there exists a graph G = (V, E) such that E1 ⊆ E ⊆ E2 and G is P4-sparse. In this paper we present a polynomial-time algorithm for solving the Graph Sandwich Problem for P4-sparse graphs. c © 2008 Elsevier B.V. All rights reserved.

In this paper we introduce the simultaneous membership problem, defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G1 and G2, sharing some vertices X (and the corresponding induced edges), are said to be simultaneous members of graph class C, if there exist representations R1 and R2 of G1 and G2 that are “consistent” on X ....

A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets AL; AR ; SL and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to vertex of AR∪SR; no vertex of SR is adjacent to a vertex of AL∪SL. The graph sandwich problem for 1-join composition is de9ned as follows: Given a vertex set V , a forced edge set E, and a ...