4 Minimal Obstructions for Partial Representations of Interval Graphs
Maria Saumell · 2015
Abstract. Interval graphs are intersection graphs of closed intervals. A gener-alization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation speci-fying some pre-drawn intervals. We ask whether we can add the remaining in-tervals and construct an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make a par-tial representation non-extendible. This generalizes Lekkerkerker and Boland’s characterization of minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to the first polynomial-time certifying algorithm for partial representation extension. 1