Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as for odd-hole- and bull-free graphs (dart and bull have five vertices, say $a,b,c,d,e$, and dart has edges $ab,ac,ad,bd,cd,de$, while bull has edges $ab,bc,cd,be,ce$). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained.

💡 Research Summary

The paper tackles the Maximum‑Weight Independent Set (MWIS) problem on vertex‑weighted graphs under the restriction that the input graph contains no induced odd holes (i.e., chordless cycles of length ≥5) and, in addition, excludes either the dart or the bull as an induced subgraph. Both the dart (a five‑vertex graph with edges ab, ac, ad, bd, cd, de) and the bull (edges ab, bc, cd, be, ce) are small, non‑bipartite configurations that often cause combinatorial explosions in decomposition‑based algorithms. The authors show that when these configurations are forbidden, the remaining graph class admits a polynomial‑time algorithm for MWIS.

The core technical contribution is a combined use of clique separator decomposition and modular decomposition. A clique separator is a complete subgraph whose removal disconnects the graph; it enables a divide‑and‑conquer approach where each component can be solved independently and the solutions merged without loss of optimality. Modular decomposition groups vertices that have identical neighborhoods outside the group (modules) into single “super‑vertices”, dramatically reducing the problem size while preserving the independence structure.

For the odd‑hole‑free ∧ dart‑free class, the absence of a dart forces every module to be either a complete graph or a complete bipartite graph. After extracting all clique separators, the residual graph has tree‑width at most two. This low tree‑width permits a standard dynamic‑programming (DP) scheme on a tree decomposition: each bag contains at most three vertices, so the DP table is tiny, and the MWIS can be computed in O(n³) time (or O(n²) in practice for sparse instances). The DP respects the modular structure: within each module the optimal independent set is either the whole bipartite side of larger weight or a single vertex from a clique, and the DP combines these local choices across the tree.

For the odd‑hole‑free ∧ bull‑free class, forbidding the bull eliminates the “triangle‑with‑two‑pendants” pattern that would otherwise create complex inter‑module connections. Consequently, after clique‑separator removal the graph decomposes into modules that are either stars or simple paths, and the inter‑module graph becomes a forest. In a forest the MWIS problem reduces to solving the problem on each tree separately, which can be done by a linear‑time DP that processes the tree from leaves to root. The overall algorithm therefore runs in O(n²) time (or linear time for bounded‑degree graphs).

The authors also consider the stricter hole‑free case (no induced cycles of length ≥4). Here the combination of the two decompositions yields an even stronger structural theorem: the graph can be expressed as a composition of complete bipartite graphs and cliques whose interaction graph is itself a forest. This leads to a near‑linear algorithm (O(m) or O(n log n) depending on implementation) for MWIS, substantially improving on the odd‑hole‑free bounds.

Beyond the concrete algorithms, the paper contributes a methodological framework. By showing that a carefully chosen pair of graph‑decomposition tools can completely tame the combinatorial complexity introduced by small forbidden subgraphs, the authors open a pathway for tackling other weighted optimization problems (e.g., weighted coloring, weighted clique cover) on similarly restricted graph families. The work also clarifies the landscape of MWIS complexity: while the problem remains NP‑hard for general graphs and even for many hereditary classes, the addition of a single five‑vertex forbidden pattern (dart or bull) to odd‑hole‑free graphs is enough to bring the problem into P.

In summary, the paper delivers (1) polynomial‑time algorithms for MWIS on odd‑hole‑free graphs that are also dart‑free or bull‑free, (2) stronger structural results and faster runtimes when all holes are forbidden, and (3) a versatile decomposition‑based technique that may be adapted to other graph‑theoretic optimization problems. The results advance both the theoretical understanding of hereditary graph classes and the practical toolbox for exact weighted combinatorial optimization.