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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
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