Dominating Induced Matchings for P7-Free Graphs in Linear Time

Let $G$ be a finite undirected graph with edge set $E$. An edge set $E’ \subseteq E$ is an {\em induced matching} in $G$ if the pairwise distance of the edges of $E’$ in $G$ is at least two; $E’$ is {\em dominating} in $G$ if every edge $e \in E \setminus E’$ intersects some edge in $E’$. The \emph{Dominating Induced Matching Problem} (\emph{DIM}, for short) asks for the existence of an induced matching $E’$ which is also dominating in $G$; this problem is also known as the \emph{Efficient Edge Domination} Problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three. However, its complexity was open for $P_k$-free graphs for any $k \ge 5$; $P_k$ denotes a chordless path with $k$ vertices and $k-1$ edges. We show in this paper that the weighted DIM problem is solvable in linear time for $P_7$-free graphs in a robust way.

💡 Research Summary

The paper “Dominating Induced Matchings for P7‑Free Graphs in Linear Time” addresses the Weighted Dominating Induced Matching (DIM) problem, also known as the Efficient Edge Domination problem, on the class of P7‑free graphs (graphs that contain no induced chordless path on seven vertices). A dominating induced matching is a set of edges M such that (i) the edges of M are pairwise at distance at least two (i.e., M is an induced matching) and (ii) every edge of the graph is either in M or shares an endpoint with an edge of M (i.e., M dominates the edge set). The weighted version asks for a minimum‑weight such set with respect to a given edge‑weight function ω:E→ℝ.

The problem is NP‑complete in general and remains hard even for very restricted graph families (planar bipartite graphs of maximum degree three). Prior to this work, the complexity of DIM for Pk‑free graphs with k≥5 was open; only for P5‑free graphs a quadratic‑time algorithm was known via bounded clique‑width techniques. The authors close this gap for k=7 by presenting a linear‑time algorithm that is also robust: it does not require a prior guarantee that the input graph is P7‑free; the algorithm either returns an optimal DIM, proves that no DIM exists, or detects a violation of the P7‑free property.

The core of the algorithm is a deep structural analysis of P7‑free graphs that admit a DIM. The authors first identify mandatory edges: any diamond (K4 minus one edge) forces its middle edge to belong to any DIM. Such edges can be removed together with their incident vertices, and all edges at distance one from them can be assigned infinite weight, guaranteeing they will never be selected. Next, they exploit the concept of homogeneous sets (modules): a vertex set H is homogeneous if every vertex outside H is either adjacent to all of H or to none of H. In a P7‑free graph with a DIM, any homogeneous set either consists of isolated vertices or a disjoint union of edges, and its external neighbourhood must be a stable set. Using linear‑time modular decomposition, the algorithm processes each maximal homogeneous set. For modules with a single external neighbour, a dedicated subprocedure (Hom‑1‑DIM) determines whether the module forces additional mandatory edges or leads to a contradiction. This subprocedure runs in O(|H|+|N(H)|) time, and the total over all modules is linear.

After removing all mandatory edges, the remaining graph G* may still contain triangle leaf blocks (a triangle attached to the rest of the graph by a single cut‑vertex). The authors replace each such triangle by a path of length two (the Tr transformation), preserving the existence and weight of optimal DIMs while simplifying the structure. The transformation is applied only when the internal blocks of G* form a distance‑hereditary bipartite graph; in this case the transformed graph remains distance‑hereditary bipartite, a class for which a linear‑time maximum induced matching algorithm is known.

The algorithm proceeds as follows:

1. Compute all maximal homogeneous sets using modular decomposition (linear time).
2. For each homogeneous set, apply Hom‑1‑DIM to extract mandatory edges or detect infeasibility.
3. Remove all mandatory edges, obtaining G*.
4. Identify triangle leaf blocks in G* and apply the Tr transformation, yielding a graph H.
5. Test whether H is a distance‑hereditary bipartite graph; if not, the input is not P7‑free or has no DIM.
6. On H, run a linear‑time algorithm for maximum induced matching in distance‑hereditary bipartite graphs, which directly yields a minimum‑weight DIM for H.
7. Combine the matching found in H with the mandatory edges collected earlier to obtain a minimum‑weight DIM for the original graph.

Each step is shown to run in O(n+m) time, where n and m are the numbers of vertices and edges, respectively. The authors also provide linear‑time procedures to test whether a given edge set is a DIM and whether it is an induced matching, which are used as subroutines throughout the algorithm.

The paper proves several structural lemmas: any graph with a DIM must be K4‑free, W4‑free, gem‑free, and Ck‑free for k≥6; the neighbourhood of any vertex is a disjoint union of at most one star containing a P3 and isolated edges/vertices; and homogeneous sets with at least two external neighbours must be either a stable set or a matching. These lemmas underpin the correctness of the reductions and guarantee that the algorithm never discards a feasible solution.

In summary, the authors deliver the first linear‑time algorithm for the weighted Dominating Induced Matching problem on P7‑free graphs. Their method combines modular decomposition, mandatory‑edge reduction, triangle‑leaf transformation, and known linear‑time matching algorithms for distance‑hereditary bipartite graphs. The algorithm is robust, automatically handling inputs that violate the P7‑free condition, and it resolves an open complexity question for the class of Pk‑free graphs with k≥5. This contribution advances both the theoretical understanding of DIM on sparse graph classes and provides a practical tool for applications in parallel resource allocation, coding theory, and network routing where efficient edge domination is relevant.