Peakless Functions on Graphs

Abstract Let G = (V, E) be a finite connected graph, endowed with the standard graph-metric d(u, v). A real valued function f defined on V is called peakless if d(u, w) + d(w, v) = d(u, v) implies f(w) ⩽ max{f(u), f(v)} and equality holds only if f(u) = f(v) (see Busemann, (1955) for the general definition). Peakless functions inherit and generalize the properties of usual convex functions. In this paper we investigate the properties of peakless functions in graphs. We define a convexity in graphs, known in the geometry of geodesies under the name “total convexity”, which is closely related with peakless functions. Namely, totally convex sets are precisely the level sets of peakless functions. In particular, a graph has no nonconstant peakless functions if and only if it does not contain proper totally convex sets. We call such graphs peakless-prime and show that an arbitrary graph G admits a decomposition in which all members are peakless-prime. These primes are exactly the subgraphs of G on which all peakless functions of G are constant. It is interesting to notice that such a decomposition into peakless-prime subgraphs represents a common modular and simplicial decomposition of G.