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Triangulating 3-colored graphs. (English) Zbl 0779.05020
A graph is called chordal if it has no induced cycles of length four or greater. A properly colored graph $$G=(V,E)$$ with coloring $$c:V\to\{1,2,\ldots,k\}$$ is said to be $$c$$-triangulated if there exists a chordal graph $$G'=(V,E')$$, where $$E\subset E'$$ and $$c$$ is a proper coloring on $$G'$$. Given a graph $$G$$ with a proper vertex coloring, the Triangulating Colored Graph Problem (TCG) asks if $$G$$ can be $$c$$- triangulated. This problem is equivalent to a problem in evolutionary tree inference called the perfect phylogeny problem. TCG is known to be NP-complete. An $$O(n^{k+1})$$ algorithm for TCG was found, where $$n$$ is the number of vertices and $$k$$ the number of colors. This gives an $$O(n^ 4)$$ algorithm for 3-colored graphs. The main contribution of this paper is to give a linear algorithm for solving TCG on 3-colored graphs.
Reviewer: K.W.Lih (Nankang)

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C05 Trees 68R10 Graph theory (including graph drawing) in computer science 92D15 Problems related to evolution 05C75 Structural characterization of families of graphs 68W10 Parallel algorithms in computer science 92D10 Genetics and epigenetics
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