Abstract
Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the 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 a linear-time certifying algorithm for partial representation extension.
Highlights
The main motivation for graph drawing and geometric representations is finding ways to visualize some given data efficiently
We generalize the characterization of Lekkerkerker and Boland [30] to describe minimal obstructions which make partial representations non-extendible
We have described the minimal obstructions that make a partial interval representation non-extendible
Summary
The main motivation for graph drawing and geometric representations is finding ways to visualize some given data efficiently. The most famous representations are plane drawings, in which we draw a graph in the plane and we want to avoid (or minimize) crossings of edges. For certain types of graphs, intersection representations are more suitable. They represent each vertex by a geometrical object and encode the edges by intersections
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