a b s t r a c tA walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.
IntroductionConsider finite, simple and undirected graphs. V and E denote the vertex set and the edge set of the graph G, respectively. A complete set of G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. The clique family of G is denoted by C(G). The clique graph of G is the intersection graph of C(G).The clique operator, K, assigns to each graph G its clique graph which is denoted by K(G). On the other hand, say that G is a clique graph if G belongs to the image of the clique operator, i.e. if there exists a graph H such that G = K (H).Clique operator and its image were widely studied. First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I
Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, B k -EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B 4 -EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that every circular-arc graph is B 3 -EPG, and that there exist circular-arc graphs which are not B 2 -EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in a rectangle E-mail addresses: of the grid), we obtain EPR (edge intersection of path in a rectangle) representations. We may define B k -EPR graphs, k ≥ 0, the same way as B k -EPG graphs. Circulararc graphs are clearly B 4 -EPR graphs and we will show that there exist circular-arc graphs that are not B 3 -EPR graphs. We also show that normal circular-arc graphs are B 2 -EPR graphs and that there exist normal circular-arc graphs that are not B 1 -EPR graphs. Finally, we characterize B 1 -EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.