A Note on Contractible Edges in Chordal Graphs

Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower connectivity. We characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators.

💡 Research Summary

The paper investigates the problem of edge contraction in k‑connected chordal graphs, focusing on which edges can be contracted without decreasing the graph’s vertex‑connectivity. After reviewing related work on contractible edges in 3‑connected and higher‑connected graphs, the authors formalize the necessary graph‑theoretic notions: vertex connectivity κ(G), minimum cut sets γ(G), and the operation of contracting an edge e={u,v} to a new vertex z_uv. An edge is defined as contractible if κ(G.e)=κ(G); a graph in which every edge is non‑contractible is termed contraction‑critical.

The core technical tool is the tree decomposition (T,l) of a chordal graph, where each node’s label l(x) is a maximal clique and the labels satisfy the usual covering, edge‑covering, and connectivity conditions. Lemma 2 establishes a fundamental property: for any adjacent nodes x and y in T, the intersection l(x)∩l(y) is non‑empty if and only if the original graph G is connected. This lemma links the combinatorial structure of the tree decomposition directly to graph connectivity.

Building on Lemma 2, Theorem 1 characterizes the set M of all minimal vertex separators of a chordal graph as the collection of inclusion‑minimal label intersections: M = {l(x)∩l(y) | {x,y}∈E(T)} after discarding any set that properly contains another. Consequently, the minimal separators can be read off directly from the tree decomposition without additional graph searches.

The main contribution, Theorem 2, gives a precise necessary and sufficient condition for an edge e={u,v} in a k‑connected chordal graph (with at least k+2 vertices) to be contractible. Two mutually exclusive scenarios guarantee contractibility:

1. e belongs to a unique maximal clique of G. In this case e appears in the label of exactly one tree‑decomposition node, so contracting e does not affect any separator.
2. There exist adjacent nodes x and y in T such that {u,v}⊆l(x)∩l(y) and the size of this intersection exceeds k, i.e., |l(x)∩l(y)|>k. The proof shows that after contraction the intersection size drops by at most one, remaining at least k, which by Lemma 2 ensures that no new separator of size <k appears; thus κ(G.e) stays equal to k.

From Theorem 2 the authors derive several corollaries. Every edge incident to a simplicial vertex (a vertex whose neighbourhood forms a unique maximal clique) is contractible, implying that any k‑connected chordal graph contains at least 2k contractible edges. For split graphs, any edge joining the clique part K to the independent set I is contractible because it lies in a unique maximal clique; however, a regular split graph that is k‑connected and has at least k+2 vertices reduces to a complete graph, making every edge non‑contractible and the graph contraction‑critical.

The paper’s methodology highlights the power of tree‑decomposition based analysis for chordal graphs. By reducing the contractibility test to simple size checks on label intersections, the authors suggest an O(|E|) algorithmic procedure once a tree decomposition is available. Limitations include the reliance on maximal‑clique labels; the results may not directly extend to arbitrary tree decompositions where node labels are not maximal cliques. Future work could explore relaxing this requirement, designing efficient algorithms for constructing suitable decompositions, or investigating sequences of edge contractions that preserve connectivity in more general graph classes.

Overall, the work provides a clean structural characterization of contractible edges in chordal graphs, bridging classical connectivity theory with modern decomposition techniques, and opens avenues for both theoretical extensions and practical algorithms in graph manipulation and network reliability.