1,482 research outputs found
Balancedness of subclasses of circular-arc graphs
A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.Fil: Bonomo, Flavia. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Duran, Guillermo Alfredo. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Universidad de Chile; ChileFil: Safe, Martin Dario. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Wagler, Annegret Katrin. Centre National de la Recherche Scientifique; Franci
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
Minimum Cost Homomorphisms to Reflexive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)H, the {\em minimum cost homomorphism problem} for HHGc_i(u)u\in V(G)i\in V(H)kGHk. We focus on the
minimum cost homomorphism problem for {\em reflexive} digraphs HHHH has a {\em Min-Max ordering}, i.e.,
if its vertices can be linearly ordered by <i<j, s<rir, js
\in A(H)is \in A(H)jr \in A(H)H$ which does not admit a Min-Max ordering, the minimum cost
homomorphism problem is NP-complete. Thus we obtain a full dichotomy
classification of the complexity of minimum cost homomorphism problems for
reflexive digraphs
On the graph condition regarding the -inverse cover problem
In their paper titled "On -inverse covers of inverse monoids", Auinger and
Szendrei have shown that every finite inverse monoid has an -inverse cover
if and only if each finite graph admits a locally finite group variety with a
certain property. We study this property and prove that the class of graphs for
which a given group variety has the required property is closed downwards in
the minor ordering, and can therefore be described by forbidden minors. We find
these forbidden minors for all varieties of Abelian groups, thus describing the
graphs for which such a group variety satisfies the above mentioned condition
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
Letter graphs and geometric grid classes of permutations: characterization and recognition
In this paper, we reveal an intriguing relationship between two seemingly
unrelated notions: letter graphs and geometric grid classes of permutations. An
important property common for both of them is well-quasi-orderability,
implying, in a non-constructive way, a polynomial-time recognition of geometric
grid classes of permutations and -letter graphs for a fixed . However,
constructive algorithms are available only for . In this paper, we present
the first constructive polynomial-time algorithm for the recognition of
-letter graphs. It is based on a structural characterization of graphs in
this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author
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