368 research outputs found
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
The Dilworth Number of Auto-Chordal-Bipartite Graphs
The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E)
has the same color classes X and Y as B, and two vertices x in X and y in Y are
adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal
bipartite if none of its induced subgraphs is a chordless cycle with at least
six vertices. In this paper, we deal with chordal bipartite graphs whose mirror
is chordal bipartite as well; we call these graphs auto-chordal bipartite
graphs (ACB graphs for short). We describe the relationship to some known graph
classes such as interval and strongly chordal graphs and we present several
characterizations of ACB graphs. We show that ACB graphs have unbounded
Dilworth number, and we characterize ACB graphs with Dilworth number k
Feedback vertex set on chordal bipartite graphs
Let G=(A,B,E) be a bipartite graph with color classes A and B. The graph G is
chordal bipartite if G has no induced cycle of length more than four. Let
G=(V,E) be a graph. A feedback vertex set F is a set of vertices F subset V
such that G-F is a forest. The feedback vertex set problem asks for a feedback
vertex set of minimal cardinality. We show that the feedback vertex set problem
can be solved in polynomial time on chordal bipartite graphs
On some special classes of contact -VPG graphs
A graph is a -VPG graph if one can associate a path on a rectangular
grid with each vertex such that two vertices are adjacent if and only if the
corresponding paths intersect at at least one grid-point. A graph is a
contact -VPG graph if it is a -VPG graph admitting a representation
with no two paths crossing and no two paths sharing an edge of the grid. In
this paper, we present a minimal forbidden induced subgraph characterisation of
contact -VPG graphs within four special graph classes: chordal graphs,
tree-cographs, -tidy graphs and -free graphs. Moreover, we present a
polynomial-time algorithm for recognising chordal contact -VPG graphs.Comment: 34 pages, 15 figure
Maximum matching width: new characterizations and a fast algorithm for dominating set
We give alternative definitions for maximum matching width, e.g. a graph
has if and only if it is a subgraph of a chordal
graph and for every maximal clique of there exists with and such that any subset of
that is a minimal separator of is a subset of either or .
Treewidth and branchwidth have alternative definitions through intersections of
subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We
show that mm-width combines both aspects, focusing on nodes and on edges. Based
on this we prove that given a graph and a branch decomposition of mm-width
we can solve Dominating Set in time , thereby beating
whenever . Note that and these inequalities are
tight. Given only the graph and using the best known algorithms to find
decompositions, maximum matching width will be better for solving Dominating
Set whenever
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