Chordal graphs and join spaces (Q583230)

A join space is composed of a set J and a join operation \(\circ\) in J subject to certain axioms reflecting the fundamental properties of line segments in Euclidean space. The join operation \(\circ\) is a mapping from \(J\times J\) to the family of subsets of J. It means that two endpoints determine a segment. An ``inverse'' operation / is defined by \(a/b=\{x|\) \(a\in b\circ x\}\). The axioms are given as follows: (1) \(a\circ b\neq \emptyset\); (2) \(a\circ b=b\circ a\); (3) \((a\circ b)\circ c=a\circ (b\circ c)\); (4) if (a/b)\(\cap (c/e)\neq \emptyset\) then (a\(\circ e)\cap (c\circ b)\neq \emptyset\); (5) a/b\(\neq \emptyset.\) An undirected simple graph is chordal if every cycle of at least four lines contains a chord. For two vertices a and b, define \(a\cdot b\) to be the shortest path between a and b. In this paper, the author proved that a cordal graph G is a join space with respect to the operation `\(\cdot '\) iff G does not contain two special graphs as induced subgraphs.