Independent Sets of Maximum Weight in Apple-Free Graphs

Abstract

Abstract. We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free.

Maximum independent sets in subcubic graphs: New results

“…• 6 does not belong to C . Indeed, if 6 belongs to C , then it must be of distance at most 2 from a (Claim 1), in which case we obtain an induced cycle of length at least p − 2 containing four edges (0, 3), (3,6), (0, 2), (2,5) • u is adjacent to 2. Indeed, assume u is not adjacent to 2.…”

“…. )-free graphs generalizes both chordal graphs and claw-free graphs, and a solution for the maximum independent set problem in this class was presented in [3]. In case of bounded degree graphs this solution can be extended to graphs without large apples, i.e.…”

“…If a is not adjacent to 3 and is adjacent to 6, then -if 6 ∈ C , i.e. if 6 = b, then 3 has no other neighbours on C by Claim 1, in which case we obtain an induced cycle of length p − 1 containing four edges (0, 3), (3,6), (0, 2), (2, 5) of H . -if 6 / ∈ C , then any other neighbour of 6 on C (if any) must be of distance at most 3 from a by Claim 1, in which case we obtain an induced cycle of length at least p − 2 containing four edges (0, 3), (3,6), (0, 2), (2,5) of H .…”

“…Case 2.8: Assume x is connected to b, y exists and it has degree 2 (that is, y is connected only to 3,5). We then apply total struction with H = G[{ y, 2, 6}] and = 3, which replaces [H] = G[{x, y, 1, 2, 3, 5, 6}] with a new vertex z adjacent to a, b, 4 and decreases α(G) by 3.…”

Weighted independent sets in a subclass of P6-free graphs

“…Let K m,n denote the complete bipartite graph with m vertices in one partition set and n vertices in the other. The banner is also called P or 4-apple or A 4 in various papers [3,5,7], and see Figure 1 for some of the special graphs that we have used in this paper.…”