1,446 research outputs found
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
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . In this paper, we are
interested in the enumeration of (inclusion-wise) minimal dominating sets in
graphs, called the Dom-Enum problem. It is well known that this problem can be
polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the
problem of enumerating all minimal transversals in a hypergraph. Firstly we
show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum
problem. As a consequence there exists an output-polynomial time algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
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
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
- …