Small (2,s)-colorable graphs without 1-obstacle representations

An obstacle representation of a graph G is a set of points on the plane together with a set of polygonal obstacles that determine a visibility graph isomorphic to G. The obstacle number of G is the minimum number of obstacles over all obstacle representations of G. Alpert, Koch, and Laison [1] gave a 12-vertex bipartite graph and proved that its obstacle number is two. We show that a 10-vertex induced subgraph of this graph has obstacle number two. Alpert et al. [1] also constructed very large graphs with vertex set consisting of a clique and an independent set in order to show that obstacle number is an unbounded parameter. We specify a 70-vertex graph with vertex set consisting of a clique and an independent set, and prove that it has obstacle number greater than one. This is an ancillary document to our article in press [8]. We conclude by showing that a 10-vertex graph with vertex set consisting of two cliques has obstacle number greater than one, improving on a result therein.