Obstructions to partitions of chordal graphs

Matrix partitions of split graphs

“…Even when restricted to chordal graphs, there are matrices for which there are infinitely many chordal minimal obstructions [1,6]. One of these matrices and an infinite family of chordal minimal obstructions to this matrix, appear frequently in relation to other classes of graphs in this paper, and so are listed in Figure 1.1.…”

Join colourings of chordal graphs

“…It has been studied in [18], where it is shown that for chordal graphs the M 1 -partition problem has only finitely many minimal obstructions, given in Fig. 1.…”

“…In fact in these two cases there is a unique chordal minimal obstruction, K k , respectively K ℓ [12], and, more generally, if M is a matrix in which all off-diagonal entries are * , then the unique chordal minimal obstruction is (ℓ+1)K k+1 [15]. Several other special cases of matrices M with polynomial M-partition problems, and finite sets of minimal obstructions, restricted to the class of chordal graphs are known [15,18,9] and are discussed in the survey [14]. However, even for chordal graphs, it is not known which matrices M have polynomial M-partition problems, nor which matrices M have only finitely many minimal chordal obstructions.…”

“…We begin by observing that there are symmetric matrices M without off-diagonal 0's that have infinitely many chordal minimal obstructions, for instance the following matrix discussed in [6,18]:…”

Minimal obstructions for a matrix partition problem in chordal graphs

“…Many papers deal with matrix partition problems for chordal graphs, e.g., in [9], a forbidden subgraph characterization and a polynomial time recognition algorithm is given for chordal graphs admitting a partition into k independent sets and cliques, [5] considers the list version of the problem on chordal graphs, polarity of chordal graphs is studied in [3], in [10] the M -partition problem is studied on chordal graphs for a special family of patterns called joining matrices. In particular, this work might be thought as a complement of [6], where both the characterization problem and the complexity problem is studied for small matrices (m × m with m < 5) on chordal graphs. In particular, they show that if M is a matrix of size m < 4, then M has finitely many chordal minimal obstructions, except for the following two matrices, which have inifinitely many chordal minimal obstructions.…”