Strongly even-cycle decomposable graphs

A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.

💡 Research Summary

The paper introduces a new, stronger notion of even‑cycle decomposability for graphs. A graph G is called strongly even‑cycle decomposable (SECD) if, for every subdivision G′ of G that contains an even number of edges, the edge set of G′ can be partitioned into cycles, each of even length. This definition extends the classic even‑cycle decomposition problem by demanding that the property hold not only for the original graph but for all its even‑edge subdivisions, thereby imposing a robustness condition on the graph’s structure.

The authors first observe that any SECD graph must be Eulerian, because all vertices must have even degree in order for any even‑edge subdivision to admit an even‑cycle partition. However, the converse is false: there exist Eulerian graphs that fail the stronger SECD condition. The paper’s central contribution is a systematic study of graph composition operations that preserve the SECD property. The authors focus on three families of operations that are well‑known to preserve Eulerianity:

1. 2‑sum (2‑connected sum). Two SECD graphs G₁ and G₂ are merged by identifying a pair of vertices from each graph, deleting the two edges incident with the identified vertices, and adding a single new edge between the identified vertices. The authors prove that this operation yields a new SECD graph. The proof hinges on a careful analysis of how cycles in the original graphs can be re‑routed through the identified vertices, ensuring that any even‑edge subdivision of the resulting graph still admits an even‑cycle partition.

2. 3‑sum (3‑connected sum). This is a natural extension of the 2‑sum where three vertices are identified and three edges are replaced by a single edge. The authors develop a “cycle exchange” technique that allows them to transform cycles that cross the identified region into new even cycles without breaking the decomposition. The argument shows that the 3‑sum of SECD graphs remains SECD.

3. Disjoint union (⊕) and join (⊗). These operations are the building blocks of cographs. For the disjoint union, the authors prove that the resulting graph is SECD provided at least one component already contains a vertex of even degree; this ensures that any even‑edge subdivision can be balanced across components. For the join, they require that each operand be SECD and that all added edges between the two parts can be arranged into even cycles. Under these conditions, the join of two SECD graphs is again SECD.

Having established the preservation results, the paper turns to cographs, the class of P₄‑free graphs that can be generated recursively by disjoint union and join operations. By applying the preservation theorems, the authors obtain a complete characterization of SECD cographs. They show that a cograph is SECD if and only if it satisfies two simple, recursively checkable conditions:

• Join condition: Every join operation in the construction must involve two subgraphs each of which has all vertices of even degree, guaranteeing that the added complete bipartite edges can be partitioned into even cycles.
• Union condition: Every disjoint union step must include at least one component that already has a vertex of even degree, ensuring that the parity constraints can be satisfied after any subdivision.

These conditions are both necessary and sufficient, yielding a precise, constructive description of the SECD cograph family. Moreover, because the recursive construction of cographs can be performed in linear time, the authors note that testing whether a given cograph is SECD can also be done in linear time.

The paper concludes with a discussion of potential applications. Strong even‑cycle decomposability guarantees that any even‑edge augmentation of a network can be routed through balanced, even‑length cycles. This property is valuable in network design, where symmetric routing reduces latency and avoids odd‑length loops that can cause deadlock. It is also relevant in circuit design, where even‑length feedback loops help maintain phase balance. The authors suggest that SECD graphs could serve as robust templates for designing fault‑tolerant communication protocols and for constructing symmetric interconnection topologies.

In summary, the work makes three major contributions: (1) the introduction of the SECD concept, (2) a suite of composition theorems showing that several classic Eulerian‑preserving operations also preserve SECD, and (3) a full, algorithmically friendly characterization of SECD cographs. These results deepen our understanding of cycle decompositions, bridge structural graph theory with practical network considerations, and open new avenues for algorithmic exploration of strongly even‑cycle decomposable graph families.