Max point-tolerance graphs

A graph G is a max point-tolerance (MPT) graph if each vertex v of G can be mapped to a pointed-interval (I-v, P-v) where I-v is an interval of R and p(v) is an element of I-v such that uv is an edge of G iff I-u boolean AND I-v superset of {p(u) p(v)}. MPT graphs model relationships among DNA fragments in genomewide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders.on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs. (C) 2015 Elsevier B.V. All rights reserved