论文信息 - Convex sets in graphs, II. Minimal path convexity

Convex sets in graphs, II. Minimal path convexity

A set K of vertices in a connected graph is M-convex if and only if for every pair of vertices in K, all vertices of all chordless paths joining them also lie in K. Caratheodory, Helly and Radon type theorems are proved for M-convex sets. The Caratheodory number is 1 for complete graphs and 2 for other graphs. The Helly number equals the size of a maximum clique. The Radon number is one more than the Helly number except possibly for triangle-free graphs, where it is at most 4.

Pierre Duchet | P. Duchet

[1] Robert E. Jamison-Waldner. PARTITION NUMBERS FOR TREES AND ORDERED SETS , 1981 .

[2] Desmond Fearnley-Sander,et al. Universal Algebra , 1982 .

[3] Jürgen Eckhoff,et al. Radon’s theorem revisited , 1979 .

[4] Robert E. Jamison,et al. A Helly theorem for convexity in graphs , 1984, Discret. Math..

[5] Pierre Duchet. Convexity in combinatorial structures , 1987 .

[6] M. Farber,et al. Convexity in graphs and hypergraphs , 1986 .

[7] R. Webster,et al. Convexity spaces. I. The basic properties , 1972 .

[8] H. M. Mulder. The interval function of a graph , 1980 .

[9] George Gratzer,et al. Universal Algebra , 1979 .

[10] P. Duchet,et al. Sous Les Pavés , 1983 .

[11] Phillip Wayne Bean. HELLY AND RADON-TYPE THEOREMS IN INTERVAL CONVEXITY SPACES , 1974 .

[12] Henry Meyniel,et al. Ensemble Convexes dans les Graphes I: Théorèmes de Helly et de Radon pour Graphes et Surfaces , 1983, Eur. J. Comb..

[13] J. Calder. Some Elementary Properties of Interval Convexities , 1971 .

[14] D. C. Kay,et al. Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers , 1971 .

[15] Jürgen Schmidt,et al. ber die Rolle der transfiniten Schluweisen in einer allgemeinen Idealtheorie , 1952 .

[16] Paul H. Edelman,et al. The theory of convex geometries , 1985 .

[17] F. W. Levi,et al. On Helly's Theorem and the Axioms of Convexity , 1951 .