Volume 30, No 1, 2023, P. 85-109

UDC 519.178
S. V. Sorochan
New cases of polynomial solvability of the independent set problem for graphs with forbidden triods

Abstract:
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:
1) a triode and all its spanning supergraphs having bounded vertex
degrees are forbidden;
2) a triode and all its spanning supergraphs having large deficit
(the number of edges in the complementary graph) are forbidden;
3) 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, triode, supergraph, polynomial algorithm.

DOI: 10.33048/daio.2023.30.752

Sergey V. Sorochan ¹
1. Lobachevsky Nizhny Novgorod State University,
23 Gagarin Avenue, 603950 Nizhny Novgorod, Russia
e-mail: sergey.sorochan@itmm.unn.ru

Received August 31, 2022
Revised November 3, 2022
Accepted November 3, 2022

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