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A faster algorithm to recognize undirected path graphs. (English) Zbl 0770.68096
Summary: Let $$\mathcal F$$ be a finite family of nonempty sets. The undirected graph $$G$$ is called the intersection graph of $$\mathcal F$$ if there is a bijection between the members of $$\mathcal F$$ and the vertices of $$G$$ such that any two sets $$F_ i$$ and $$F_ j$$ (for $$i \neq j$$) have a nonempty intersection if and only if the corresponding vertices are adjacent. We study intersection graphs where $$\mathcal F$$ is a family of undirected paths in an unrooted, undirected tree; these graphs are called (undirected) path graphs. They constitute a proper subclass of the chordal graphs. F. Gavril [Discrete Math. 23, 211-227 (1978; Zbl 0398.05060)] gave the first polynomial time algorithm to recognize undirected path graphs; his algorithm runs in time $$O(n^ 4)$$, where $$n$$ is the number of vertices. The topic of this paper is a new recognition algorithm that runs in time $$O(mn)$$, where $$m$$ is the number of edges.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity 05C05 Trees 05C75 Structural characterization of families of graphs
##### Keywords:
intersection graph; chordal graphs; undirected path graphs
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##### References:
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