The Simultaneous Membership Problem for Chordal, Comparability and Permutation graphs

In this paper we introduce the ‘simultaneous membership problem’, defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G_1 and G_2, sharing some vertices X (and the corresponding induced edges), are said to be ‘simultaneous members’ of graph class C, if there exist representations R_1 and R_2 of G_1 and G_2 that are “consistent” on X. Equivalently (for the classes C that we consider) there exist edges E’ between G_1-X and G_2-X such that G_1 \cup G_2 \cup E’ belongs to class C. Simultaneous membership problems have application in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated version of the other. Simultaneous membership problems are related to simultaneous planar embeddings, graph sandwich problems and probe graph recognition problems. In this paper we give efficient algorithms for the simultaneous membership problem on chordal, comparability and permutation graphs. These results imply that graph sandwich problems for the above classes are tractable for an interesting special case: when the set of optional edges form a complete bipartite graph. Our results complement the recent polynomial time recognition algorithms for probe chordal, comparability, and permutation graphs, where the set of optional edges form a clique.

💡 Research Summary

The paper introduces the “simultaneous membership problem” for graph classes that admit a geometric or combinatorial representation, focusing on chordal, comparability, and permutation graphs. Given two graphs G₁ = (V₁, E₁) and G₂ = (V₂, E₂) that share a set X of vertices (and the induced edges), the question is whether there exist representations R₁ and R₂ of G₁ and G₂, respectively, that agree on X. Equivalently, the problem can be phrased as: does there exist a set A ⊆ (V₁ \ X) × (V₂ \ X) of “augmenting” edges such that the combined graph G = (V₁ ∪ V₂, E₁ ∪ E₂ ∪ A) belongs to the target class C? The authors observe that this formulation is a special case of the graph sandwich problem where the optional edges form a complete bipartite graph.

For chordal graphs, the authors develop a structural notion called an S‑elimination vertex. A vertex v is S‑eliminable if its neighbors in G₁ (N₁(v)) and its neighbors in G₂ (N₂(v)) each induce a clique in their respective graphs. Lemma 1 shows that any augmentable instance must contain such a vertex. Theorem 2 proves that removing an S‑elimination vertex v and adding all cross‑edges C(v) = { (x, y) | x ∈ N₁(v) \ X, y ∈ N₂(v) \ X } preserves augmentability. This yields a simple recursive algorithm: repeatedly find an S‑elimination vertex, add its cross‑edges, delete the vertex, and continue. The algorithm runs in O(n³) time; the O(n) test for whether a vertex is S‑eliminable uses a perfect elimination ordering of each input chordal graph to check cliquehood of neighbor sets efficiently.

For comparability graphs, the problem is expressed in terms of transitive orientations. If G₁ and G₂ admit transitive orientations T₁ and T₂ that agree on the shared edges E(X), then the union T = T₁ ∪ T₂ is a “pseudo‑transitive” orientation of the combined graph. Theorem 3 shows that any pseudo‑transitive orientation can be extended to a full transitive orientation by adding a suitable set A of augmenting edges. Consequently, the simultaneous membership problem reduces to finding A such that G₁ ∪ G₂ ∪ A is a comparability graph. The authors adapt Golumbic’s O(n m) algorithm for recognizing comparability graphs and constructing a transitive orientation, extending it to handle the two‑graph setting. The resulting algorithm decides simultaneous comparability in O(n m) time.

Permutation graphs are precisely the comparability graphs of permutations; they can be represented as the intersection of two linear orders without crossing. Because the comparability‑graph algorithm already works for any graph whose edges can be oriented transitively, it directly applies to permutation graphs, yielding the same O(n m) bound.

The paper also discusses the relationship to probe graph recognition, where the optional edges form a clique, and to simultaneous planar embeddings, highlighting that the simultaneous membership problem is a natural graph‑theoretic analogue of those geometric problems. By restricting the optional edges to a complete bipartite structure, the authors identify a tractable subclass of the otherwise NP‑complete graph sandwich problem for the three studied classes.

In summary, the authors provide:

1. A formal equivalence between simultaneous membership and a special graph sandwich problem (complete bipartite optional edges).
2. An O(n³) algorithm for chordal graphs based on recursive elimination of S‑elimination vertices.
3. An O(n m) algorithm for comparability graphs using extensions of Golumbic’s transitive‑orientation technique.
4. The same O(n m) bound for permutation graphs via the comparability‑graph reduction. These results broaden the algorithmic toolkit for handling temporally or evolutionarily related graph data, with potential applications in computational biology (e.g., DNA fragment overlap graphs), social network analysis, and dynamic graph visualization.