Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Global outer connected domination number of a graph

In this paper we obtain some bounds for outer connected domination numbers and global outer connected domination numbers of graphs

Revolutionaries and spies: Spy-good and spy-bad graphs

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if $m$ of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever. Let $\sigma(G,m,r)$ denote the minimum number of spies needed to win. To avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the lower bound is sharp when $G$ has a rooted spanning tree $T$ such that every edge of $G$ not in $T$ joins two vertices having the same parent in $T$. As a consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where $\gamma(G)$ is the domination number; this bound is nearly sharp when $\gamma(G)\le m$. For the random graph with constant edge-probability $p$, we obtain constants $c$ and $c'$ (depending on $m$ and $p$) such that $\sigma(G,m,r)$ is near the trivial upper bound when $r<c\ln n$ and at most $c'$ times the trivial lower bound when $r>c'\ln n$. For the hypercube $Q_d$ with $d\ge r$, we have $\sigma(G,m,r)=r-m+1$ when $m=2$, and for $m\ge 3$ at least $r-39m$ spies are needed. For complete $k$-partite graphs with partite sets of size at least $2r$, the leading term in $\sigma(G,m,r)$ is approximately $\frac{k}{k-1}\frac{r}{m}$ when $k\ge m$. For $k=2$, we have \sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and \sigma(G,3,r)=\floor{r/2}, and in general $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.Comment: 34 pages, 2 figures. The most important changes in this revision are improvements of the results on hypercubes and random graphs. The proof of the previous hypercube result has been deleted, but the statement remains because it is stronger for m<52. In the random graph section we added a spy-strategy resul

Rainbow Colorings in Graphs

In this thesis, we deal with rainbow colorings of graphs. We engage not with the rainbow connection number but with counting of rainbow colorings in graphs with k colors. We introduce the rainbow polynomial and prove some results for some special graph classes. Furthermore, we obtain bounds for the rainbow polynomial. In addition, we define some edge colorings related to the rainbow coloring, like the s-rainbow coloring and the 2-rainbow coloring. For this edge colorings, polynomials are defined and we prove some basic properties for this polynomials and present some formulas for the calculation in special graph classes. In addition, we consider in this thesis counting problems related to the rainbow coloring like rainbow pairs and rainbow dependent sets. We introduce polynomials for this counting problems and present some general properties and formulas for special graph classes

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Advances and Novel Approaches in Discrete Optimization

Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms