42,758 research outputs found
The leafage of a chordal graph
The leafage l(G) of a chordal graph G is the minimum number of leaves of a
tree in which G has an intersection representation by subtrees. We obtain upper
and lower bounds on l(G) and compute it on special classes. The maximum of l(G)
on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G)
is the minimum number of leaves when no subtree may contain another; we obtain
upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free
chordal graphs. We use asteroidal sets and structural properties of chordal
graphs.Comment: 19 pages, 3 figure
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
Bounded Representations of Interval and Proper Interval Graphs
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of
recognition called the bounded representation problem which we study for the
classes of interval and proper interval graphs. The input gives a graph G and
in addition for each vertex v two intervals L_v and R_v called bounds. We ask
whether there exists a bounded representation in which each interval I_v has
its left endpoint in L_v and its right endpoint in R_v. We show that the
problem can be solved in linear time for interval graphs and in quadratic time
for proper interval graphs.
Robert's Theorem states that the classes of proper interval graphs and unit
interval graphs are equal. Surprisingly the bounded representation problem is
polynomially solvable for proper interval graphs and NP-complete for unit
interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the
proper and unit interval representations behave very differently.
The bounded representation problem belongs to a wider class of restricted
representation problems. These problems are generalizations of the
well-understood recognition problem, and they ask whether there exists a
representation of G satisfying some additional constraints. The bounded
representation problems generalize many of these problems
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
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