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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.

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
Full Text: DOI
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