Geodesic
New cases of polynomial solvability of~the~independent set problem for~graphs~with~forbidden~triods
Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 1, pp. 85-109.

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A triode is a tree with three leaves and a single vertex of degree $3$. The independent set problem is solvable in polynomial time for graphs that do not contain a triode as a subgraph with any fixed number of vertices. If the induced triode having $k$ vertices is forbidden, then for $k>5$ the complexity of this problem is unknown. We consider intermediate cases when an induced triode with any fixed number of vertices and some of its spanning supergraphs are forbidden. For an arbitrary triode with a fixed vertex number, we prove the solvability of the independent set problem in polynomial time in the following cases: a triode and all its spanning supergraphs having bounded vertex degrees are forbidden; a triode and all its spanning supergraphs having large deficit (the number of edges in the complementary graph) are forbidden; a triode and all its supergraphs from which this triode can be obtained using the graph intersection operation are forbidden, provided the fraph has a vertex with bounded anti-degree. Bibliogr. 20.
Keywords: independent set, IS-easy class, IS-hard class, monotonic class, hereditary class, forbidden subgraph, supergraph
Mots-clés : triode, polynomial algorithm. 
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S. V. Sorochan. New cases of polynomial solvability of~the~independent set problem for~graphs~with~forbidden~triods. Diskretnyj analiz i issledovanie operacij, Tome 30 (2023) no. 1, pp. 85-109. http://geodesic.mathdoc.fr/item/DA_2023_30_1_a4/

[1] V. E. Alekseev and D. V. Korobitsyn, “Complexity of some problems on hereditary classes of graphs”, Diskretn. Mat., 4:4 (1992), 34–40 (Russian) | MR | Zbl

[2] V. E. Alekseev and S. V. Sorochan, “New cases of the polynomial solvability of the independent set problem for graphs with forbidden paths”, J. Appl. Ind. Math., 12:2 (2018), 213–219 | DOI | MR | Zbl

[3] V. E. Alekseev, “The local restriction effect on the complexity of finding the graph independence number”, Combinatorial and Algebraic Methods in Applied Mathematics, Izd-vo Gorkov. Univ., Gorkiy, 1982, 3–13 | MR

[4] V. E. Alekseev, “The number of maximal independent sets in graphs from hereditary classes”, Combinatorial and Algebraic Methods in Discrete Optimization, Izd-vo NNGU, Nizhny Novgorod, 1991, 5–8

[5] V. E. Alekseev, “A polynomial algorithm for finding the largest independent sets in claw-free graphs”, Diskretn. Anal. Issled. Oper., Ser. 1, 6:4 (1999), 3–19 (Russian) | Zbl

[6] Lozin V. V., Mosca R., “Independent sets in extension of $2K_2$-free graphs”, Discrete Appl. Math., 146 (2005), 74–80 | DOI | MR | Zbl

[7] Lokshantov D., Vatshelle M., Villanger Y., “Independent set in $P_5$-free graphs in polynomial time”, Proc. 25th Annual ACM-SIAM Symp. Discrete Algorithms (Portland, OR, USA, Jan. 5–7, 2014), SIAM, Philadelphia, PA, 2014, 570–581 | DOI | MR | Zbl

[8] Grzesik A., Klimošová T., Pilipczuk Mar., Pilipczuk Mic., Polynomial-time algorithm for maximum weight independent set on $P_6$-free graphs, Cornell Univ. Libr. e-Print Archive, Cornell Univ, Ithaca, NY, 2017, arXiv: 1707.05491 | MR

[9] Karthick T., Maffray F., “Weighted independent sets in classes of $P_6$-free graphs”, Discrete Appl. Math., 209 (2016), 217–226 | DOI | MR | Zbl

[10] Lozin V. V., Monnot J., Ries B., “On the maximum independent set problem in subclasses of subcubic graphs”, J. Discrete Algorithms, 31 (2015), 104–112 | DOI | MR | Zbl

[11] Lozin V. V., Rautenbach D., “Some results on graphs without long induced paths”, Inform. Process. Lett., 88 (2003), 167–171 | DOI | MR | Zbl

[12] D. S. Malyshev and D. V. Sirotkin, “Polynomial-time solvability of the independent set problem in a certain class of subcubic planar graphs”, J. Appl. Ind. Math., 11:3 (2017), 400–414 | DOI | MR | Zbl

[13] Abrishami T., Chudnovsky M., Dibek C., Rzążewski P., “Polynomial-time algorithm for maximum independent set in bounded-degree graphs with no long induced claws”, Proc. 33rd Annual ACM-SIAM Symp. Discrete Algorithms (Alexandria, VA, USA, Jan. 9–12, 2022), SIAM, Philadelphia, PA, 2022, 1448–1470 | MR

[14] Alekseev V. E., Lozin V. V., Malyshev D. S., Milanič M., “The maximum independent set problem in planar graphs”, Mathematical foundations of computer science 2008, Proc. 33rd Int. Symp. (Toruń, Poland, Aug. 25–29 2008), Lect. Notes Comput. Sci., 5162, Springer, Heidelberg, 2008, 96–107 | DOI | MR | Zbl

[15] D. S. Malyshev, “Classes of subcubic planar graphs for which the independent set problem is polynomially solvable”, J. Appl. Ind. Math., 7:4 (2013), 537–548 | DOI | MR | Zbl

[16] Lovász L., Plummer M. D., Matching theory, Ann. Discrete Math., 29, Norh-Holland, Amsterdam, 1986, 544 pp. | MR | Zbl

[17] Minty G., “On maximal independent sets of vertices in claw-free graphs”, J. Comb. Theory. Ser. B, 28:3 (1980), 284–304 | DOI | MR | Zbl

[18] Sbihi N., “Algorithme de recherche d'un stable de cardinalite maximum dans un graphe sans etoile”, Discrete Math., 29:1 (1980), 53–76 (French) | DOI | MR | Zbl

[19] V. A. Emelichev, O. I. Melnikov, V. I. Sarvanov, and R. I. Tyshkevich, Lectures on Graph Theory, B. I. Wissenschaftsverlag, Mannheim, 1994 | MR | MR | Zbl

[20] Hsu W., Ikura Y., Nemhauser G. L., “A polynomial algorithm for maximum weighted vertex packing in graphs without long odd cycles”, Math. Program., 20 (1981), 225–232 | DOI | MR | Zbl