20 research outputs found
Graph isomorphism completeness for trapezoid graphs
The complexity of the graph isomorphism problem for trapezoid graphs has been
open over a decade. This paper shows that the problem is GI-complete. More
precisely, we show that the graph isomorphism problem is GI-complete for
comparability graphs of partially ordered sets with interval dimension 2 and
height 3. In contrast, the problem is known to be solvable in polynomial time
for comparability graphs of partially ordered sets with interval dimension at
most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure
Stage-graph representations
AbstractWe consider graph applications of the well-known paradigm “killing two birds with one stone”. In the plane, this gives rise to a stage graph as follows: vertices are the points, and u, v is an edge if and only if the (infinite, straight) line segment joining u to v intersects the stage. Such graphs are shown to be comparability graphs of ordered sets of dimension 2. Similar graphs can be constructed when we have a fixed number k of stages on the plane. In this case, u, v is an edge if and only if the (straight) line segment uv intersects one of the k stages. In this paper, we study stage representations of stage graphs and give upper and lower bounds on the number of stages needed to represent a graph
A polynomial-time algorithm for the paired-domination problem on permutation graphs
Author name used in this publication: T. C. E. Cheng2008-2009 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe
Fair allocation of indivisible goods under conflict constraints
We consider the fair allocation of indivisible items to several agents and
add a graph theoretical perspective to this classical problem. Thereby we
introduce an incompatibility relation between pairs of items described in terms
of a conflict graph. Every subset of items assigned to one agent has to form an
independent set in this graph. Thus, the allocation of items to the agents
corresponds to a partial coloring of the conflict graph. Every agent has its
own profit valuation for every item. Aiming at a fair allocation, our goal is
the maximization of the lowest total profit of items allocated to any one of
the agents. The resulting optimization problem contains, as special cases, both
{\sc Partition} and {\sc Independent Set}. In our contribution we derive
complexity and algorithmic results depending on the properties of the given
graph. We can show that the problem is strongly NP-hard for bipartite graphs
and their line graphs, and solvable in pseudo-polynomial time for the classes
of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and
graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also
be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here
appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix
with a remark about biconvex bipartite graph
The Firebreak Problem
Suppose we have a network that is represented by a graph . Potentially a
fire (or other type of contagion) might erupt at some vertex of . We are
able to respond to this outbreak by establishing a firebreak at other
vertices of , so that the fire cannot pass through these fortified vertices.
The question that now arises is which vertices will result in the greatest
number of vertices being saved from the fire, assuming that the fire will
spread to every vertex that is not fully behind the vertices of the
firebreak. This is the essence of the {\sc Firebreak} decision problem, which
is the focus of this paper. We establish that the problem is intractable on the
class of split graphs as well as on the class of bipartite graphs, but can be
solved in linear time when restricted to graphs having constant-bounded
treewidth, or in polynomial time when restricted to intersection graphs. We
also consider some closely related problems
The interval inclusion number of a partially ordered set
AbstractA containment representation for a poset P is a map ƒ such that x<y in P if and only if ƒ(x) ⊂ ƒ(y). We introduce the interval inclusion number (or interval number) i(P) as the smallest t such that P has a containment representation f in which each f(x) is the union of at most t intervals. Trivially, i(P)=1 if and only if dim(P)⩽2. Posets with i(P)=2 include the standard n-dimensional poset and all interval orders; i.e. posets of arbitrarily high dimension. In general we have the upper bound i(P) ⩽ ⌈;dim(P)2⌉, with equality holding for the Boolean algebras. For lexicographic composition, i(P) = k and dim(Q) = 2k + 1 imply i(P[Q]) = k + 1. This result and i(B2k) = k imply that testing i(P) ⩽ k for any fixed k ⩾ 2 is NP-complete. Concerning removal theorems, we prove that i(P − x) ⩾ i(P) − 1 when x is a maximal or minimal element of P, and in general i(P − x) ⩾ i(P)2