Computer Science

Let G=(V,E) be a simple, undirected and connected graph. A connected (total) dominating set S⊆V is a secure connected (total) dominating set of G , if for each u∈V∖S , there exists v∈S such that uv∈E and (S∖{v})∪{u} is a connected (total) dominating set of G . The minimum cardinality of a secure connected (total) dominating set of G denoted by γ sc (G)( γ st (G)) , is called the secure connected (total) domination number of G . In this paper, we show that the decision problems corresponding to secure connected domination number and secure total domination number are NP-complete even when restricted to split graphs or bipartite graphs. The NP-complete reductions also show that these problems are w[2]-hard. We also prove that the secure connected domination problem is linear time solvable in block graphs and threshold graphs.

Taking the covering dimension dim as notion for the dimension of a topological space, we first specify thenumber zdim_{T_0}(n) of zero-dimensional T_0-spaces on {1,...,n}$ and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,\ldots,n} by means oftwo mappings po and P that yieldthe number po(n) of partial orders on {1,...,n} and the set P(n) of partitions of {1,...,n}, respectively. Algorithms for both mappings exist. Assuming one for po to be at hand, we use our specification of zdim_{T_0}(n) and modify one for P in such a way that it computes zdim_{T_0}(n) instead of P(n). The specification of zdim(n) then allows to compute this number from zdim_{T_0}(1) to zdim_{T_0}(n) and the Stirling numbers of the second kind S(n,1) to S(n,n). The resulting algorithms have been implemented in C and we also present results of practical experiments with them. To considerably reduce the running times for computing zdim_{T_0}(n), we also describe a backtracking approach and its parallel implementation in C using the OpenMP library.

We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a \emph{geodetic set} if every vertex of G lies in a shortest path between some pair of vertices of S . The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study \textsc{MGS} on restricted classes of planar graphs: we design a linear-time algorithm for \textsc{MGS} on solid grids, improving on a 3 -approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that it remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that \textsc{MGS} is fixed parameter tractable for inputs of this class when parameterized by its \emph{tree-width} (which equals its clique number). This implies a polynomial-time algorithm for k -trees, for fixed k . Then, we show that \textsc{MGS} is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure.

Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p≥3 both variants of the problem become NP-complete when restricted to split ( S 3 ,…, S p ) -free graphs, where S q denotes the q -sun graph.

A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in linear time. We further show that `5' is tight for every wheel graph with at least 5 vertices, and briefly discuss possible generalizations to planar graphs of treewidth 3.

We present a novel algorithm for the minimum-depth elimination tree problem, which is equivalent to the optimal treedepth decomposition problem. Our algorithm makes use of two cheaply-computed lower bound functions to prune the search tree, along with symmetry-breaking and domination rules. We present an empirical study showing that the algorithm outperforms the current state-of-the-art solver (which is based on a SAT encoding) by orders of magnitude on a range of graph classes.

Partitioning a graph using graph separators, and particularly clique separators, are well-known techniques to decompose a graph into smaller units which can be treated independently. It was previously known that the treewidth was bounded above by the sum of the size of the separator plus the treewidth of disjoint components, and this was obtained by the heuristic of filling in all edges of the separator making it into a clique. In this paper, we present a new, tighter upper bound on the treewidth of a graph obtained by only partially filling in the edges of a separator. In particular, the method completes just those pairs of separator vertices that are adjacent to a common component, and indicates a more effective heuristic than filling in the entire separator. We discuss the relevance of this result for combinatorial algorithms and give an example of how the tighter bound can be exploited in the domain of constraint satisfaction problems.

We give an account of Agafonov's original proof of his eponymous theorem. The original proof was only reported in Russian in a journal not widely available, and the work most commonly cited in western literature is instead the english translation of a summary version containing no proofs. The account contains some embellishments to Agafonov's original arguments, made in the interest of clarity, and provides some historical context to Agafonov's work.

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on n nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let L be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with L ∗ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that L≤ L ∗ +1.5D , where D is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with L= L ∗ +1.01D . Recently, Skutella [Sku16] improved these bounds by showing that L≤ L ∗ + 19 14 D , and there exists an instance with L= L ∗ +1.1D . We contribute to this line of research by showing that L≤ L ∗ +1.3D . We also take a first step towards lower and upper bounds for small instances.

A set $U\subseteq \reals^2$ is n -universal if all n -vertex planar graphs have a planar straight-line embedding into U . We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then Q must have cardinality Ω( n 2 ) in order to be n -universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an o( n 2 ) upper bound on universal sets.