Convexities Related to Path Properties on Graphs; a Unified Approach

Path properties, such as ’geodesic’, ’induced’, ’all paths’ define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Carathéodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties. ∗The research for this paper was mainly done while this author was visiting the Econometric Institute of Erasmus University, Rotterdam and the University of Groningen as a BOYSCAST fellow of the Department of Science and Technology (DST) of the Ministry of Science and Technology of India in 1998. The financial support of the DST, New Delhi, and the University of Groningen, and the hospitality of the Econometric Institute, Rotterdam, are greatly acknowledged.