A Structural Characterization for Certifying Robinsonian Matrices

Abstract

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian.

Perfect elimination orderings for symmetric matrices

“…In other words, this extends to matrices the well known fact that unit interval graphs are interval graphs, which in turn are chordal and cocomparability graphs. As shown in [19] the structural characterization of unit interval graphs in terms of minimal forbidden structures extends naturally to the matrix setting and in this paper (Theorem 1) we provide such an extension for chordal graphs. Establishing such extensions for interval and cocomparability matrices, or a more general theory for generalizing structural characterizations from graphs to matrices, is an interesting open problem which we leave for further research.…”

“…There is a well known structural characterization of unit interval graphs in terms of minimal forbidden substructures (namely, claws and asteroidal triples; see [13,22]). An analogous structural characterization was given in [19] for Robinsonian matrices (by extending the notion of asteroidal triple to weighted graphs). For chordal graphs the minimal forbidden substructures are the chordless cycles.…”

On the Recognition of Strong-Robinsonian Incomplete Matrices

“…Later in [15], the same authors presented a recognition algorithm with time complexity O(n 2 + nm log n) that uses similarity first search. Again, using the relationship between Robinsonian matrices and unit interval graphs, Laurent et al in [16] gave a characterization of Robinsonian matrices via forbidden patterns.…”

Modules in Robinson Spaces

“…Subsequently, two new recognition algorithms were proposed by Laurent and Seminaroti: in [25] they presented an algorithm of complexity O(α⋅n) based on classical LexBFS traversal and divide-and-conquer (where α is the depth of the recursion tree, which is at most the number of distinct elements of the input matrix), and in [26] they presented an O(n 2 log n) algorithm, which extends LexBFS to weighted matrices and is used as a multisweep traversal. Laurent, Seminaroti and Tanigawa [27] presented a characterization of Robinson matrices in terms of forbidden substructures, extending the notion of asteroidal triples in graphs to weighted graphs. More recently, Aracena and Thraves Caro [1] presented a parametrized algorithm for the NP-complete problem of recognition of Robinson incomplete matrices.…”