16,539 research outputs found
Powers of cycles, powers of paths, and distance graphs
In 1988, Golumbic and Hammer characterized powers of cycles, relating them to circular-arc graphs. We extend their results and propose several further structural characterizations for both powers of cycles and powers of paths. The characterizations lead to linear-time recognition algorithms of these classes of graphs. Furthermore, as a generalization of powers of cycles, powers of paths, and even of the well-known circulant graphs, we consider distance graphs. While colourings of these graphs have been intensively studied, the recognition problem has been so far neglected. We propose polynomial-time recognition algorithms for these graphs under additional restrictions
On Intersection Graphs of Arcs and Chords in a Circle
Circular-arc graphs are the intersection graphs of arcs on a circle. We review in this thesis the main results known about this class and we analize some subclasses of it. We show new characterizations for proper circular-arc graphs derived from a characterization formulated by Tucker, and we deduce minimal forbidden structures for circular arc-graphs.All possible intersections of the defined subclasses are studied, showing a minimal example in each one of the generated regions, except one of them that we prove it is empty. From here, we conclude that a clique-Helly and proper no unit circular-arc graph must be Hellycircular-arc graph.Circle graphs are the intersection graphs of chords in a circle. We present also a review of the main results in this class and define the most important subclasses, proving some relations of inclusions between them.We prove a neccesary condition so that a graph is a Helly circle graph and conjecture thatthis condition is sufficient too. If this conjecture becomes true, we would have acharacterization and a polynomial recognition for this subclass.Minimal forbidden structures for circle graphs are shown, using the chacterization of propercircular-arc graphs by Tucker and a characterization theorem for circle graphs by Bouchet.We also analize all the possible intersections between the defined subclasses of circlegraphs, showing a minimal example in each generated region.A superclass of circle graphs is studied: overlap graphs of circular-arc graphs. We show new properties on this class, analizing its relation with circle and circular-arc graphs. A necessary condition for a graph being an overlap graph of circular-arc graphs is shown. We prove that the problem of finding a minimum clique partition for the class of graphs which does not contain either odd holes, or a 3-fan, or a 4-wheel as induced subgraphs, can be solved in polynomial time. We use in the proof results of polyhedral theory for integer linear programming. We extend this result for minimum clique covering by vertices. These results are applied for Helly circle graphs without odd holes. We also show that the problem of minimum clique covering by vertices can be solved in polynomial time for Helly circular-arc graphs. Finally, we present some interesting problems which remain open.Sociedad Argentina de Informática e Investigación Operativ
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
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
On the structure of (pan, even hole)-free graphs
A hole is a chordless cycle with at least four vertices. A pan is a graph
which consists of a hole and a single vertex with precisely one neighbor on the
hole. An even hole is a hole with an even number of vertices. We prove that a
(pan, even hole)-free graph can be decomposed by clique cutsets into
essentially unit circular-arc graphs. This structure theorem is the basis of
our -time certifying algorithm for recognizing (pan, even hole)-free
graphs and for our -time algorithm to optimally color them.
Using this structure theorem, we show that the tree-width of a (pan, even
hole)-free graph is at most 1.5 times the clique number minus 1, and thus the
chromatic number is at most 1.5 times the clique number.Comment: Accepted to appear in the Journal of Graph Theor
Isomorphism of graph classes related to the circular-ones property
We give a linear-time algorithm that checks for isomorphism between two 0-1
matrices that obey the circular-ones property. This algorithm leads to
linear-time isomorphism algorithms for related graph classes, including Helly
circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and
convex-round graphs.Comment: 25 pages, 9 figure
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