A mixed multigraph is a multigraph which may contain both undirected and directed edges. An orientation of a mixed multigraph $G$ is an assignment of exactly one direction to each undirected edge of $G$. A mixed multigraph $G$ can be oriented to a strongly connected digraph if and only if $G$ is bridgeless and strongly connected [Boesch and Tindell, Am. Math. Mon., 1980]. For each $r \in \mathbb{N}$, let $f(r)$ denote the smallest number such that any strongly connected bridgeless mixed multigraph with radius $r$ can be oriented to a digraph of radius at most $f(r)$. We improve the current best upper bound of $4r^2+4r$ on $f(r)$ [Chung, Garey and Tarjan, Networks, 1985] to $1.5 r^2 + r + 1$. Our upper bound is tight upto a multiplicative factor of $1.5$ since, $\forall r \in \mathbb{N}$, there exists an undirected bridgeless graph of radius $r$ such that every orientation of it has radius at least $r^2 + r$ [Chv\'atal and Thomassen, J. Comb. Theory. Ser. B., 1978]. We prove a marginally better lower bound, $f(r) \geq r^2 + 3r + 1$, for mixed multigraphs. While this marginal improvement does not help with asymptotic estimates, it clears a natural suspicion that, like undirected graphs, $f(r)$ may be equal to $r^2 + r$ even for mixed multigraphs. En route, we show that if each edge of $G$ lies in a cycle of length at most $\eta$, then the oriented radius of $G$ is at most $1.5 r \eta$. All our proofs are constructive and lend themselves to polynomial time algorithms.
翻译:混合的多面体是一个多面体, 它可能包含非方向和定向的边际。 混合的多面体$G$的定向是每个非方向边缘的精确方向。 混合的多面体$G$如果而且只有$G美元没有桥梁,而且[Boesch和Tindell, Am. Math. M., 1980], 才能适应一个紧密相连的字形。 对于每个$( $ ) 和 mathbb{N} 的多面体。 我们的上界值是最小的数, 因为任何与半径2美元紧密相连的平面混合体$( $美元) 和美元。 混合的G$2+4r2+4r$( r) 我们的上界值是1.5 r2 + r 。 我们的上界值最接近于1.5美元, 我们的上界值是最低的平面值 。