# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Beeri, C.; Fagin, R.; Maier, D.; Yannakakis, M., On the desirability of acyclic database schemes, J. ACM, 30, 479-513, (1983) · Zbl 0624.68087  Booth, K.S.; Lueker, G.S., Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. comput. system sci., 13, 335-379, (1976) · Zbl 0367.68034  Buneman, P., A characterization of rigid circuit graphs, Discrete math., 9, 205-212, (1974) · Zbl 0288.05128  Dietz, P.F., Intersection graph algorithms, TR 84-628 (ph.D. thesis), (1984), Computer Science Department, Cornell University Ithaca, NY  Fagin, R., Degrees of acyclicity for hypergraphs and relational database schemes, J. ACM, 30, 514-550, (1983) · Zbl 0624.68088  Gavril, F., Algorithms for minimum coloring, maximum clique, minimum covering by cliques and maximum independent set of a chordal graph, SIAM J. comput., 1, 180-187, (1972) · Zbl 0227.05116  Gavril, F., The intersection graphs of subtrees in trees are exactly the chordal graphs, J. combin. theory ser. B, 16, 47-56, (1974) · Zbl 0266.05101  Gavril, F., A recognition algorithm for the intersection graphs of directed paths in directed trees, Discrete math., 13, 237-249, (1975) · Zbl 0312.05108  Gavril, F., A recognition algorithm for the intersection graphs of paths in trees, Discrete math., 23, 211-227, (1978) · Zbl 0398.05060  Gilmore, P.C.; Hoffman, A.J., A characterization of comparability graphs and interval graphs, Canad. J. math., 16, 539-548, (1964) · Zbl 0121.26003  Golumbic, M.C., Algorithmic graph theory and perfect graphs, (1980), Academic Press New York · Zbl 0541.05054  Johnson, D.S., The NP-completeness column: an ongoing guide, J. algorithms, 6, 434-451, (1985) · Zbl 0608.68032  Monma, C.L.; Wei, V.K., Intersection graphs of paths in a tree, J. combin. theory ser. B, 41, 141-181, (1986) · Zbl 0595.05062  Novick, M.B., Parallel algorithms for intersection graphs, TR 90-1096 (ph.D. thesis), (1990), Computer Science Department, Cornell University Ithaca, NY  Renz, P.L., Intersection representations of graphs by arcs, Pacific J. math., 34, 501-510, (1970) · Zbl 0191.55103  Rose, D.J., Triangulated graphs and the elimination process, J. math. anal. appl., 32, 597-609, (1970) · Zbl 0216.02602  Rose, D.J., A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, (), 183-217  Rose, D.J.; Tarjan, R.E.; Lueker, G.J., Algorithmic aspects of vertex elimination on graphs, SIAM J. comput., 5, 266-283, (1976) · Zbl 0353.65019  Tarjan, R.E.; Yannakakis, M., Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs, SIAM J. comput., 13, 566-579, (1984) · Zbl 0545.68062  Yannakakis, M.; Gavril, F., The maximum k-colorable subgraph problem for chordal graphs, Inform. process. lett., 24, 133-137, (1987) · Zbl 0653.68070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.