Degree-Constrained Orientations of Embedded Graphs

Abstract

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This primal-dual orientation problem was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by 2 2g , where g is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes NP-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.

A Connection Between Sports and Matroids: How Many Teams Can We Beat?

“…Since then, various extensions of the problem have been examined, see e.g. [5,7,10,11]. Asahiro et al [2] considered a variant of the degree-constrained orientation problem where a penalty function on the violated degree bounds is to be minimized, but to the best of our knowledge vdibo as defined here has not been studied before.…”

The Minimum Feasible Tileset Problem

“…For example, Biedl et al [2] proposed an approximation algorithm for finding a balanced acyclic orientation. Another natural constraint on an orientation that has been studied is to prescribe degrees for each vertex [10,13,17].…”

“…Specifically, symbols whose pattern contains some fixed I ⊆ [k] are assigned to tiles that contain I in their pattern. By constraint (10) the number of symbols and the number of tiles are equal. Note that each tile is used for two disjoint sets I, J ⊆ [k] and each variable x I,J appears in two (10)-constraints (for I and for J).…”

“…By constraint (10) the number of symbols and the number of tiles are equal. Note that each tile is used for two disjoint sets I, J ⊆ [k] and each variable x I,J appears in two (10)-constraints (for I and for J). Thus, each tile with pattern {I, J} is assigned two symbols, one requiring the tile for the scenarios in I and the other requiring it the ones in J.…”

Flips on homologous orientations of surface graphs with prescribed forbidden facial circuits

“…Graph orientation with certain degree-constraints on vertices is a natural focus of study in graph theory and has close connection with many combinatorial structures in graphs, such as the primal-dual orientations [1], spanning trees [2], bipolar orientations [4], transversal structures [5], Schnyder woods [6], bipartite perfect matchings (or more generally, bipartite ffactors) [8,10,12] and c-orientations of the dual of plane graph [9,12]. In general, all these constraints could be encoded in terms of α-orientations [2], that is, the orientations with prescribed out-degrees.…”