Improved bounds for the oriented radius of mixed multigraphs

Abstract

AbstractA mixed multigraph is an ordered pair with a set of vertices, a multiset of unordered and ordered pairs of vertices, respectively, called undirected and directed edges. An orientation of a mixed multigraph is an assignment of exactly one direction to each undirected edge of . A mixed multigraph can be oriented to a strongly connected digraph if and only if is bridgeless and strongly connected. For each , let denote the smallest number such that any strongly connected bridgeless mixed multigraph with radius can be oriented to a digraph of radius at most . We improve the current best upper bound of on  to . Our upper bound is tight up to a multiplicative factor of 1.5 since, for every positive integer , there exists an undirected bridgeless graph of radius such that every orientation of it has a radius at least . We prove a marginally better lower bound, , for mixed multigraphs. While this marginal improvement does not help with asymptotic estimates, it clears a natural suspicion that, like, undirected graphs, may be equal to even for mixed multigraphs. En route, we show that if each edge of lies in a cycle of length at most , then the oriented radius of is at most . All our proofs are constructive and lend themselves to polynomial‐time algorithms.