Weighted Well-Covered Graphs without Cycles of Length 4, 5, 6 and 7

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove that finding the vector space of weight functions under which an input graph is w-well-covered can be done in polynomial time, if the input graph does not contain cycles of length 4, 5, 6 and 7.

💡 Research Summary

The paper investigates weighted well‑covered graphs, a natural extension of the classic well‑covered concept. A graph G is well‑covered if every maximal independent set (MIS) has the same cardinality; it is w‑well‑covered with respect to a weight function w : V→ℝ if every MIS has the same total weight. The collection of all such weight functions forms a vector space, but determining this space for an arbitrary graph is computationally hard. The authors focus on a restricted class of simple graphs that contain no cycles of length 4, 5, 6, or 7 (i.e., they are C₄‑, C₅‑, C₆‑, C₇‑free). This structural restriction yields powerful combinatorial properties that the authors exploit.

The central notion introduced is that of a generating subgraph. A generating subgraph B is an induced complete bipartite subgraph with bipartition (B_X, B_Y) such that there exists an independent set S with the property that both S∪B_X and S∪B_Y are maximal independent sets of G. Consequently, any weight function w that makes G w‑well‑covered must satisfy the linear constraint w(B_X)=w(B_Y). Proposition 2.1 characterizes generating subgraphs: B is generating iff there exists an independent set inside the second‑neighbourhood N₂(B) that dominates the symmetric difference N₁(B_X)△N₁(B_Y). This condition reduces the problem to a domination question in a well‑structured neighbourhood.

The authors prove (Theorem 2.2) that for C₄‑, C₆‑, C₇‑free graphs, deciding whether a given induced complete bipartite subgraph B is generating can be performed in polynomial time. The key observation is that, because C₄ is forbidden, the smaller side B_X must consist of a single vertex x. The larger side B_Y = {y₁,…,y_k} is examined through layered neighbourhoods: M₁(y)=N₁(y)∩N₂(x) and M₂(y)=N₂(y)∩N₃(x). The absence of short cycles imposes strict independence and adjacency constraints on these layers (e.g., distinct y_i share only x as a common neighbor, M₂ sets are pairwise non‑adjacent, etc.). Using these properties, the authors construct a flow network F_P (for P∈{B_X,B_Y}) whose vertices are M₁(P), M₂(P), auxiliary nodes representing connected components of M₂(P), a source s and a sink t. All edges have unit capacity. A maximum flow is computed via the classic Ford‑Fulkerson (or Edmonds‑Karp) algorithm. The set of M₂(P) vertices carrying positive flow forms an independent set that dominates M₁(P). If such a set exists for both sides, B is generating; otherwise it is not. The algorithm runs in O(|V|·(|V|+|E|)) time, because each augmentation adds at least one unit of flow and there are at most |V| augmentations.

To connect generating subgraphs with weighted well‑coveredness, the paper leverages the theory of hereditary systems. A weighted hereditary system (H,w) is greedy if all maximal feasible sets have equal weight. The independent sets of a graph constitute the feasible family, so (G,w) is greedy exactly when G is w‑well‑covered. Theorem 3.1 (from prior work) provides a characterization of non‑greedy systems via exchange properties. Applying this to graphs yields Theorem 3.2: G fails to be w‑well‑covered iff there exist two MIS S₁, S₂ with different weights such that the induced subgraph on S₁△S₂ is a complete bipartite graph. Consequently, Theorem 3.3 establishes the equivalence: G is w‑well‑covered ⇔ G satisfies all linear constraints generated by its generating subgraphs. Thus, the vector space of admissible weight functions is precisely the intersection of the solution spaces of these constraints.

The final technical contribution (Theorem 4.1) shows how to compute this vector space efficiently. For each vertex v, define L_v as the subspace of weight functions satisfying all constraints from generating subgraphs whose smaller bipartition side is {v}. The overall admissible space is ⋂_{v∈V} L_v. To obtain L_v, the algorithm enumerates all neighbours y∈N₁(v), constructs the corresponding M₁(y) and M₂(y) layers, and uses the flow‑based test from Theorem 2.2 to decide whether y belongs to the set D(v) of “related” neighbours. Each connected component of D(v) contributes at most one vertex to a candidate generating subgraph B*. By building a family F_v consisting of such B* and its derived subgraphs, and applying the flow test to each, the algorithm extracts all linear equations that define L_v. Since each flow computation is polynomial and the number of vertices is |V|, the total runtime remains O(|V|·(|V|+|E|)).

In summary, the paper delivers a polynomial‑time algorithm for determining the full vector space of weight functions that render a C₄‑, C₅‑, C₆‑, C₇‑free graph w‑well‑covered. This resolves, for this graph class, a problem that is co‑NP‑complete in general. The methodology blends structural graph theory (absence of short cycles), combinatorial domination, flow network reductions, and hereditary system theory. The authors also highlight open directions, notably the complexity of the well‑covered recognition problem for graphs that forbid only C₄ (the original open problem posed by Brown, Nowakowski, and Zverovich). Their work suggests that further restrictions on cycle lengths or additional graph parameters may yield tractable cases, and that the generating‑subgraph/flow framework could be adapted to other weighted graph invariants.