New Cases of Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Triodes

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 with \( k \) vertices is forbidden, then for \( k>5 \) the complexity of this problem is unknown. We consider intermediate cases where 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 with bounded vertex degrees are forbidden. 2. A triode and all its spanning supergraphs having large deficiency (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 graph has a vertex of bounded antidegree.