Characterizations of probe interval graphs

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bipartite graph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3.

💡 Research Summary

The paper investigates structural properties of probe interval graphs (PIGs) from the perspective of their adjacency matrices and provides algorithmic tools for handling related graph classes. After recalling that an interval graph is defined by intersecting intervals on the real line, the authors introduce probe interval graphs as a generalization where the vertex set is partitioned into probes P and non‑probes N; two vertices are adjacent if their intervals intersect and at least one endpoint belongs to P. This definition immediately shows that PIGs strictly contain interval graphs (e.g., C₄ is a PIG but not an interval graph) and are contained in the intersection of interval split graphs and interval k‑graphs, though none of these inclusions are reversible.

The first major contribution is Observation 1.4, which gives a matrix‑based characterization of PIGs. By adding 1’s on the diagonal of the ordinary adjacency matrix one obtains the augmented adjacency matrix A(G). The graph is a PIG with probe set P = V \ N (where N is any independent set) if and only if there exists a simultaneous permutation of rows and columns that makes A(G) satisfy the “quasi‑x‑linear‑ones” property. This property extends the classic quasi‑linear‑ones condition for interval graphs: every 0 to the right of the diagonal may contain only 0 or a special symbol X (corresponding to non‑probe vertices), and similarly for zeros below the diagonal. Thus the probe/non‑probe distinction is encoded directly in the matrix pattern, providing a simple, testable criterion.

The second contribution concerns interval bipartite graphs (IBGs). For a bipartite graph B = (X, Y, E) the bi‑adjacency matrix A can be given an R‑C partition, where each zero is marked either R (right‑type) or C (down‑type) according to a stair‑shaped partition. The authors propose a diagonalization process: insert auxiliary rows and columns (filled with a placeholder X except for a 1 on the diagonal) so that every R lies strictly to the right of the main diagonal and every C lies strictly below it. The resulting square matrix ˜A is called the diagonalized form. Algorithm 2.3 then scans each row and column of ˜A to locate the last 1, defining intervals