2,581 research outputs found
On The b-Chromatic Number of Regular Graphs Without 4-Cycle
The b-chromatic number of a graph , denoted by , is the largest
integer that admits a proper -coloring such that each color class
has a vertex that is adjacent to at least one vertex in each of the other color
classes. We prove that for each -regular graph which contains no
4-cycle, and if has a triangle,
then . Also, if is a -regular
graph which contains no 4-cycle and , then .
Finally, we show that for any -regular graph which does not contain
4-cycle and ,
The effect of girth on the kernelization complexity of Connected Dominating Set
In the Connected Dominating Set problem we are given as input a graph and a positive integer , and are asked if there is a set of at most vertices of such that is a dominating set of and the subgraph induced by is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set.
Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function . The new instance is called a kernel for the problem. If is a polynomial in then we say that the problem admits polynomial kernels. The girth of a graph is the length of a shortest cycle in . It turns out that Connected Dominating Set is ``hard\u27\u27 on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth or (since the problem is W[2]-hard); (b) admits a kernel, where is , on graphs of girth or but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic () kernel on graphs of girth at least .
While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
On the Connectivity of Token Graphs of Trees
Let and be integers such that , and let be a
simple graph of order . The -token graph of is the graph
whose vertices are the -subsets of , where two vertices are adjacent
in whenever their symmetric difference is an edge of . In this
paper we show that if is a tree, then the connectivity of is equal
to the minimum degree of
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