Proper Orientations of Chordal Graphs
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An orientation D of a graph G = (V, E) is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each v ∈ V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u) ≠ dD−(v), for all uv ∈ E(G). An orientation with maximum indegree at most k is called a k-orientation. The proper orientation number of G, denoted by χ→(G), is the minimum integer k such that G admits a proper k-orientation. We prove that determining whether χ→(G) ≤ k is NP-complete for chordal graphs of bounded diameter. We also present a tight upper bound for χ→(G) on split graphs and a linear-time algorithm for quasi-threshold graphs.
1992 ◽
Vol 43
(6)
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pp. 297-300
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2005 ◽
Vol DMTCS Proceedings vol. AE,...
(Proceedings)
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2001 ◽
Vol 2
(1)
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pp. 1-23
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2009 ◽
Vol 110
(1)
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pp. 20-23
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2021 ◽
Vol vol. 23 no. 1
(Discrete Algorithms)
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