Proper Orientations of Chordal Graphs

  • Julio Araujo UFC
  • Alexandre Cezar UFC
  • Carlos Vinícius Gomes Costa Lima UFC
  • Vinicius Fernandes dos Santos UFMG
  • Ana Shirley Ferreira Silva UFC


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.

Palavras-chave: Graph Theory, Coloring, Orientation, Computational Complexity, Chordal graphs


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ARAUJO, Julio; CEZAR, Alexandre; LIMA, Carlos Vinícius Gomes Costa; DOS SANTOS, Vinicius Fernandes; SILVA, Ana Shirley Ferreira. Proper Orientations of Chordal Graphs. In: ENCONTRO DE TEORIA DA COMPUTAÇÃO (ETC), 5. , 2020, Cuiabá. Anais [...]. Porto Alegre: Sociedade Brasileira de Computação, 2020 . p. 21-24. ISSN 2595-6116. DOI: