Steiner intervals in graphs Original Research ArticleSteiner intervals in graphs Original Research Article

LetG be a graph andu, v two vertices ofG. Then the interval fromu tov consists of all those vertices that lie on some shortestu — v path. LetS be a set of vertices in a connected graphG. Then the Steiner distancedG(S) ofS inG is the smallest number of edges in a connected subgraph ofG that containsS. Such a subgraph is necessarily a tree called a Steiner tree forS. The Steiner intervalIG(S) ofS consists of all those vertices that lie on some Steiner tree forS. LetS be ann-set of vertices ofG and suppose thatk ⩽ n. Then thek-intersection interval of S, denoted byIk(S) is the intersection of all Steiner intervals of allk-subsets ofS. It is shown that ifS = {u1, u2, ..., un} is a set ofn ⩾ 2 vertices of a graphG and if the 2-intersection interval ofS is nonempty andxεI2(S), then.d(S) = ∑ni = 1 =1 d(ui, x). It is observed that the only graphs for which the 2-intersection intervals of alln-sets,n ⩾ 4, are nonempty are stars. Moreover, for everyn ⩾ 4, those graphs with the property that the 3-intersection interval of everyn-set is nonempty are completely characterized. In general, ifn = 2k, those graphsG for whichIk(S) is nonempty for everyn-setS ofG are characterized.