30,923 research outputs found
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Minimal Obstructions for Partial Representations of Interval Graphs
Interval graphs are intersection graphs of closed intervals. A generalization
of recognition called partial representation extension was introduced recently.
The input gives an interval graph with a partial representation specifying some
pre-drawn intervals. We ask whether the remaining intervals can be added to
create an extending representation. Two linear-time algorithms are known for
solving this problem.
In this paper, we characterize the minimal obstructions which make partial
representations non-extendible. This generalizes Lekkerkerker and Boland's
characterization of the minimal forbidden induced subgraphs of interval graphs.
Each minimal obstruction consists of a forbidden induced subgraph together with
at most four pre-drawn intervals. A Helly-type result follows: A partial
representation is extendible if and only if every quadruple of pre-drawn
intervals is extendible by itself. Our characterization leads to a linear-time
certifying algorithm for partial representation extension
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
Simultaneous Representation of Proper and Unit Interval Graphs
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary
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