Computer Science

A picture-hanging puzzle is the task of hanging a framed picture with a wire around a set of nails in such a way that it can remain hanging on certain specified sets of nails, but will fall if any more are removed. The classical brain teaser asks us to hang a picture on two nails in such a way that it falls when any one is detached. Demaine et al (2012) proved that all reasonable puzzles of this kind are solvable, and that for the k -out-of- n problem, the size of a solution can be bounded by a polynomial in n . We give simplified proofs of these facts, for the latter leading to a reasonable exponent in the polynomial bound.

The classical, linear-time solvable Feedback Edge Set problem is concerned with finding a minimum number of edges intersecting all cycles in a (static, unweighted) graph. We provide a first study of this problem in the setting of temporal graphs, where edges are present only at certain points in time. We find that there are four natural generalizations of Feedback Edge Set, all of which turn out to be NP-hard. We also study the tractability of these problems with respect to several parameters (solution size, lifetime, and number of graph vertices, among others) and obtain some parameterized hardness but also fixed-parameter tractability results.

In this paper, we study the feedback game on 3 -chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every 3 -chromatic Eulerian triangulation of a surface all of whose vertices have degree 0 modulo 4 is always fixed. Moreover, we also study the case of 3 -chromatic Eulerian triangulations of surfaces which have at least two vertices whose degrees are 2 modulo 4 , and in particular, we determine the winner of the game on a concrete class of such graphs, called an octahedral path.

In this paper, some subclasses of block graphs are considered in order to analyze Fiedler vector of its members. Two families of block graphs with cliques of fixed size, the block-path and block-starlike graphs, are introduced. Cases A and B of classification for both families were considered, as well as the behavior of the algebraic connectivity for particular cases of block-path graphs.

Multicriteria Decision Making problems are important both for individuals and groups. Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. We focus on decision problems where the set of pairwise comparisons can be chosen, i.e., it is not given a priori. The objective of this paper is to provide recommendations for filling patterns of incomplete pairwise comparison matrices (PCMs) based on their graph representation. Regularity means that each item is compared to others for the same number of times, resulting in a kind of symmetry. A graph on an odd number of vertices is called quasi-regular, if the degree of every vertex is the same odd number, except for one vertex whose degree is larger by one. If there is a pair of items such that their shortest connecting path is very long, the comparison between these two items relies on many intermediate comparisons, and is possibly biased by all of their errors. Such an example was previously found, where the graph generated from the table tennis players' matches included a long shortest path between two vertices (players), and the calculated result appeared to be misleading. If the diameter of the graph of comparisons is low as possible (among the graphs of the same number of edges), we can avoid, or, at least decrease, such cumulated errors. The aim of our research is to find graphs, among regular and quasi-regular ones, with minimal diameter. Both theorists and practitioners can use the results, given in several formats in the appendix: graph, adjacency matrix, list of edges.

Let G=(V,E) be a finite undirected graph. An edge subset E ′ ⊆E is a {\em dominating induced matching} ({\em d.i.m.}) in G if every edge in E is intersected by exactly one edge of E ′ . The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in G . The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but was solved in linear time for P 7 -free graphs and in polynomial time for P 8 -free graphs. In this paper, we solve it in polynomial time for P 9 -free graphs.

Let G=(V,E) be a finite undirected graph. An edge set E ′ ⊆E is a {\em dominating induced matching} ({\em d.i.m.}) in G if every edge in E is intersected by exactly one edge of E ′ . The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in G ; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for P 7 -free graphs, and in polynomial time for S 1,2,4 -free graphs as well as for S 2,2,2 -free graphs and for S 2,2,3 -free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for S 1,1,5 -free graphs.

A vertex set D in a finite undirected graph G is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of G if every vertex of G is dominated by exactly one vertex of D . The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in G , is known to be \NP-complete for P 7 -free graphs, and even for very restricted H -free bipartite graph classes such as for K 1,4 -free bipartite graphs as well as for C 4 -free bipartite graphs while it is solvable in polynomial time for P 7 -free bipartite graphs as well as for S 2,2,4 -free bipartite graphs. Here we show that ED can be solved in polynomial time for P 8 -free bipartite graphs.

A vertex set D in a finite undirected graph G is an {\em efficient dominating set} (e.d.s.\ for short) of G if every vertex of G is dominated by exactly one vertex of D . The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in G , is \NP-complete for various H -free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least g for every fixed g . Thus, ED is \NP-complete for K 1,4 -free bipartite graphs and for C 4 -free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for S 1,1,5 -free bipartite graphs.

A vertex set D in a finite undirected graph G is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of G if every vertex of G is dominated by exactly one vertex of D . The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in G , is \NP-complete for various H -free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least g for every fixed g . Thus, ED is \NP-complete for K 1,4 -free bipartite graphs and for C 4 -free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for S 1,3,3 -free bipartite graphs.