We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given any graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices is in FPT parameterized by k on well-partitioned chordal graphs, while on chordal graphs, these problems are only known to be in XP. From the other end, we observe that there are problems that are polynomial-time solvable on split graphs, but become NP-complete on well-partitioned chordal graphs.
If $M$ is an $m \times m$ matrix over $\{ 0, 1, \ast \}$, an $M$-partition of a graph $G$ is a partition $(V_1, \dots V_m)$ such that $V_i$ is completely adjacent (non-adjacent) to $V_j$ if $M_{ij} = 1$ ($M_{ij} = 0$), and there are no further restrictions between $V_i$ and $V_j$ if $M_{ij} = \ast$. Having an $M$-partition is a hereditary property, thus it can be characterized by a set of minimal obstructions (forbidden induced subgraphs minimal with the property of not having an $M$-partition). It is known that for every $3 \times 3$ matrix $M$ over $\{ 0, 1, \ast \}$, there are finitely many…
A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland. We show that a circular-arc graph cannot have a blocking quadruple. We also observe that the absence of blocking quadruples is not in general sufficient to guarantee that a graph is a circ…
We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to…
Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal ob…
Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.
For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge …
A symmetric $m\times m$ matrix $M$ with entries taken from $\{0,1,\ast\}$ gives rise to a graph partition problem, asking whether a graph can be partitioned into $m$ vertex sets matched to the rows (and corresponding columns) of $M$ such that, if $M_{ij}=1$, then any two vertices between the corresponding vertex sets are joined by an edge, and if $M_{ij}=0$ then any two vertices between the corresponding vertex sets are not joined by an edge. The entry $\ast$ places no restriction on the edges between the corresponding sets. This problem generalises graph colouring and graph homomorphism probl…
A matching $M$ in a graph $G$ is $r$-degenerate if the subgraph of $G$ induced by the set of vertices incident with an edge in $M$ is $r$-degenerate. Goddard, Hedetniemi, Hedetniemi, and Laskar (Generalized subgraph-restricted matchings in graphs, Discrete Mathematics 293 (2005) 129-138) introduced the notion of acyclic matchings, which coincide with $1$-degenerate matchings. Solving a problem they posed, we describe an efficient algorithm to determine the maximum size of an $r$-degenerate matching in a given chordal graph. Furthermore, we study the $r$-chromatic index of a graph defined as th…
Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed…
We study the perfect phylogeny problem and establish a generalization of the four gamete condition (also called the Splits Equivalence Theorem) for sequences over three state characters. Our main result is that a set of input sequences over three state characters allows a perfect phylogeny if and only if every subset of three characters allows a perfect phylogeny. In establishing these results, we prove fundamental structural features of the perfect phylogeny problem on three state characters and completely characterize the minimal obstruction sets that must occur in three state input sequence…
University of Liverpool（利物浦大学）；The Czech Academy of Sciences, Institute of Computer Science（捷克科学院计算机研究所）；Charles University, Computer Science Institute（查理大学计算机研究所）
