Separation axiom S3 for geodesic convexity in graphs
Résumé
Semispaces of a convexity space (X, C) are maximal convex sets missing a point. The separation axiom
S3 asserts that any point x0 ∈ X and any convex set A not containing x0 can be separated by complementary
halfspaces (convex sets with convex complements) or, equivalently, that all semispaces are halfspaces. The character-
ization of semispaces and the S3-axiom in linear spaces are classical results in convexity. In this paper, we study S3
for geodesic convexity in graphs and the structure of semispaces in S3-graphs. We characterize S3-graphs and their
semispaces in terms of separation by halfspaces of vertices x0 and special sets, called maximal x0-proximal sets and in
terms of convexity of their mutual shadows x0/K and K/x0. In S3-graphs G satisfying the triangle condition (TC),
maximal proximal sets are the pre-maximal cliques of G (i.e., cliques K such that K ∪{x0} are maximal cliques). This
allows to characterize the S3-graphs satisfying (TC) in a structural way and to enumerate their semispaces efficiently.
In case of meshed graphs (an important subclass of graphs satisfying (TC)), the S3-graphs have been characterized
by excluding five forbidden subgraphs. On the way of proving this result, we also establish some properties of meshed
graphs, which maybe of independent interest. In particular, we show that any connected, locally-convex set of a
meshed graph is convex. We also provide several examples of S3-graphs, including the basis graphs of matroids.
Finally, we consider the (NP-complete) halfspace separation problem, describe two methods of its solution, and apply
them to particular classes of graphs and graph-convexities.
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