Parameterized Complexity of Secluded Connectivity Problems

The Secluded Path problem models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. The problems were introduced by Chechik et al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded Path is fixed-parameter tractable (FPT) on unweighted graphs being parameterized by the maximum vertex degree of the graph and that Secluded Steiner Tree is FPT parameterized by the treewidth of the graph. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems. We give FPT-algorithms deciding if a graph G with a given cost function contains a secluded path and a secluded Steiner tree of exposure at most k with the cost at most C. We initiate the study of "above guarantee" parameterizations for secluded problems, where the lower bound is given by the size of a Steiner tree. We investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.Comment: Minor corrections are don

Editing to a Graph of Given Degrees

We consider the Editing to a Graph of Given Degrees problem that asks for a graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d}, whether it is possible to obtain a graph G' from G such that the degree of v is \delta(v) for any vertex v by at most k vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by d+k. We complement this result by showing that the problem has no polynomial kernel unless NP\subseteq coNP/poly

Computing the Chromatic Number Using Graph Decompositions via Matrix Rank

Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying $q$-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators. We present two algorithms for $q$-Coloring parameterized by cutwidth $cutw$: a deterministic one that runs in time $O^*(2^{\omega \cdot cutw})$, where $\omega$ is the matrix multiplication constant, and a randomized one with runtime $O^*(2^{cutw})$. In sharp contrast to earlier work, the running time is independent of $q$. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time assuming the Strong Exponential Time Hypothesis (SETH). Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an $O^*((\lfloor d/2\rfloor+1)^{pw})$ time randomized algorithm for $q$-Coloring on graphs of pathwidth $pw$ and maximum degree $d$. Such a runtime was first obtained by Bj\"orklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no $O^*((\lfloor d/2\rfloor+1-\varepsilon)^{pw})$-time algorithm exists assuming SETH.Comment: 29 pages. An extended abstract appears in the proceedings of the 26th Annual European Symposium on Algorithms, ESA 201

On the super connectivity of Kronecker products of graphs

In this paper we present the super connectivity of Kronecker product of a general graph and a complete graph.Comment: 8 page

Partial Colorings of Graphs

Recently, there has been great interest in counting the number of homomorphisms from a graph G into a fixed image graph H. For this thesis, we let H be a complete graph on three vertices with exactly one looped vertex. Homomorphisms from a graph G to this H correspond to partial proper two-colorings of the vertices of G. We are mainly interested in finding which graphs maximize the number of partial two-colorings given a graph with n vertices and m edges. The general result is given for all graphs with m \u3c n -1 as well as basic enumerative results for some very common graphs

Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values