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Completing orientations of partially oriented graphs
Author(s) -
BangJensen J.,
Huang J.,
Zhu X.
Publication year - 2018
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22157
Subject(s) - mathematics , combinatorics , indifference graph , transitive relation , chordal graph , pathwidth , orientation (vector space) , discrete mathematics , longest path problem , time complexity , maximal independent set , graph , modular decomposition , 1 planar graph , line graph , geometry
We initiate a general study of what we call orientation completion problems. For a fixed class C of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in C by orienting the (nonoriented) edges in P . Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs. We study orientation completion problems for various classes of oriented graphs, including k ‐arc‐strong oriented graphs, k ‐strong oriented graphs, quasi‐transitive‐oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in‐tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP‐complete. We also show that some of the NP‐complete problems become polynomial time solvable when the input‐oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.