Abstract
Penta is the configuration shown in figure 1(a), where continuous lines represent edges and dotted lines represent non-edges. The vertex u in figure 1(a) is called the center of Penta. A graph G is called a pentagraph if every induced subgraph H of G has a vertex v which is not a center of induced Penta in H. The class of pentagraphs is a common generalization of chordal [triangulated] graphs and Mahadev graphs. We construct a polynomial-time algorithm that either find a maximum stable set of G or concludes that G is not a pentagraph. We propose a method for extending α-polynomial hereditary classes based on induced Pentas.
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References
V.E. Alekseev, “A polynomial algorithm for finding the largest independent sets in claw-free graphs,” Diskretn. Anal. Issled. Oper. Ser., vol. 16, no. 4, pp. 3–19, 1999.
G.A. Dirac, “On rigid circuit graphs,” Abh. Math. Sem. Univ. Hamburg, vol. 25, pp. 71–76, 1961.
M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness, W.H. Freeman and Co., San Francisco, Calif., 1979, x+338 pp.
F. Gavril, Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph, SIAM J. Comput., vol. 1, no. 2, pp. 180–187, 1972.
P.L. Hammer and I.E. Zverovich, “Construction of maximal stable sets with k-extensions, 9 pp., Combinat., Probab. and Comput. (accepted)
F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969, ix+274 pp.
N.V.R. Mahadev, “Vertex deletion and stability number,” Techn. Report ORWP 90/2, Swiss Federal Institute of Technology, 1990.
O. Melnikov, V. Sarvanov, R. Tyshkevich, V. Yemelichev, and I. Zverovich, Exercises in Graph Theory, Kluwer Texts in the Mathematical Sciences 19, Kluwer Academic Publishers Group, Dordrecht, 1998, viii+354 pp.
G.J. Minty, On maximal independent sets of vertices in claw–free graphs, J. Combin. Theory Ser. B, vol. 28, no. 3, pp. 284–303, 1980.
D. Nakamura and A. Tamura, “A revision of Minty’s algorithm for finding a maximum weight stable set of a claw-free graph,” J. Oper. Res. Soc. Japan, vol. 44, no. 2, pp. 194–204, 2001.
D. Rautenbach, I.E. Zverovich, and I.I. Zverovich, Extensions of α-polynomial classes, Austral. J. Combin., vol. 26, pp. 265–271, 2002.
N. Sbihi, “Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoite,” Discrete Math., vol. 29, no. 1, pp. 53–76, 1980. (in French)
I.E. Zverovich, “Extension of hereditary classes with substitutions,” Discrete Appl. Math., vol. 128, nos. 2/3, pp. 487–509, 2003a.
I.E. Zverovich, Minimum degree algorithms for stability number, in: Special Issue on Graph Stability and Related Topics, Discrete Appl. Math., vol. 132, nos. 1–3, pp. 211–216, 2003.
I.E. Zverovich and I. I. Zverovich, “A characterization of superbipartite graphs,” Graph Theory Notes of New York, vol. XLVI, pp. 31–35, 2004.
I.E. Zverovich and I.I. Zverovich, “Ramseian partitions and weighted stability in graphs,” 7 pp., Bulletin Inst. Combin. Appl. (accepted).
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Zverovich, I.E., Zverovich, I.I. Penta-Extensions of Hereditary Classes of Graphs. J Comb Optim 10, 169–178 (2005). https://doi.org/10.1007/s10878-005-2271-0
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DOI: https://doi.org/10.1007/s10878-005-2271-0