2,831 research outputs found
On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can
be represented as the edge intersection graph of paths on a grid (EPG graph),
i.e., one can associate with each vertex of the graph a nontrivial path on a
rectangular grid such that two vertices are adjacent if and only if the
corresponding paths share at least one edge of the grid. For a nonnegative
integer , -EPG graphs are defined as EPG graphs admitting a model in
which each path has at most bends. Circular-arc graphs are intersection
graphs of open arcs of a circle. It is easy to see that every circular-arc
graph is a -EPG graph, by embedding the circle into a rectangle of the
grid. In this paper, we prove that every circular-arc graph is -EPG, and
that there exist circular-arc graphs which are not -EPG. If we restrict
ourselves to rectangular representations (i.e., the union of the paths used in
the model is contained in a rectangle of the grid), we obtain EPR (edge
intersection of path in a rectangle) representations. We may define -EPR
graphs, , the same way as -EPG graphs. Circular-arc graphs are
clearly -EPR graphs and we will show that there exist circular-arc graphs
that are not -EPR graphs. We also show that normal circular-arc graphs are
-EPR graphs and that there exist normal circular-arc graphs that are not
-EPR graphs. Finally, we characterize -EPR graphs by a family of
minimal forbidden induced subgraphs, and show that they form a subclass of
normal Helly circular-arc graphs
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
Subclasses of Normal Helly Circular-Arc Graphs
A Helly circular-arc model M = (C,A) is a circle C together with a Helly
family \A of arcs of C. If no arc is contained in any other, then M is a proper
Helly circular-arc model, if every arc has the same length, then M is a unit
Helly circular-arc model, and if there are no two arcs covering the circle,
then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit
Helly, normal Helly) circular-arc graph is the intersection graph of the arcs
of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model.
In this article we study these subclasses of Helly circular-arc graphs. We show
natural generalizations of several properties of (proper) interval graphs that
hold for some of these Helly circular-arc subclasses. Next, we describe
characterizations for the subclasses of Helly circular-arc graphs, including
forbidden induced subgraphs characterizations. These characterizations lead to
efficient algorithms for recognizing graphs within these classes. Finally, we
show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled
Proper Helly Circular-Arc Graphs) appeared at WG'0
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