Bandelt, H.-J.; Mulder, H. M. Helly theorems for dismantlable graphs and pseudo-modular graphs. (English) Zbl 0697.05034 Topics in combinatorics and graph theory. Essays in honour of Gerhard Ringel, 65-71 (1990). Summary: [For the entire collection see Zbl 0698.00017.] The geodesic convexity of a graph consists of all those subsets of the vertex-set which all geodesics (i.e., shortest paths) joining any two of its elements. The Helly number of this convexity is trivially bounded from below by the clique number (i.e., the size of a largest clique). We show that equality between the two numbers hold for graphs which are dismantlable (alias cop-win) or pseudo-modular. This generalizes previously known results for chordal graphs and distance-hereditary graphs, due to Čepoj, Duchet, Jamison and Nowakowski, respectively. Cited in 1 ReviewCited in 8 Documents MSC: 05C35 Extremal problems in graph theory 05C38 Paths and cycles Keywords:geodesic convexity of a graph; Helly number; clique number; chordal graphs; distance-hereditary graphs Citations:Zbl 0698.00017 PDFBibTeX XML