Blocking total dominating sets via edge contractions

An eternal $1$-secure set, in a graph $G = (V, E)$ is a set $D \subset V$ having the property that for any finite sequence of vertices $r_1, r_2, \ldots, r_k$ there exists a sequence of vertices $v_1, v_2, \ldots, v_k$ and a sequence $ D = D_0, D_1, D_2, \ldots, D_k$ of dominating sets of $G$, such that for each $i$, $1 \leq i \leq k$, $D_{i} = (D_{i-1} - \{v_i\}) \cup \{r_i\}$, where $v_i \in D_{i-1}$ and $r_i \in N[v_i]$. Here $r_i = v_i$ is possible. The cardinality of the smallest eternal $1$-secure set in a graph $G$ is called the eternal $1$-security number of $G$. In this paper we study a variations of eternal $1$-secure sets named safe eternal $1$-secure sets. A vertex $v$ is safe with respect to an eternal $1$-secure set $S$ if $N[v] \bigcap S =1$. An eternal 1 secure set $S$ is a safe eternal 1 secure set if at least one vertex in $G$ is safe with respect to the set $S$. We characterize the class of graphs having safe eternal $1$-secure sets for which all vertices - excluding those in the safe $1$-secure sets - are safe. Also we introduce a new kind of directed graphs which represent the transformation from one safe 1 - secure set to another safe 1-secure set of a given graph and study its properties.

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Modern Applications of Graph Theory

10.1093/oso/9780198856740.001.0001 ◽

2021 ◽

Author(s):

Vadim Zverovich

Keyword(s):

Graph Theory ◽

Molecular Epidemiology ◽

Emergency Response ◽

Real Life ◽

Dominating Sets ◽

Postgraduate Students ◽

Graph Theoretic ◽

Electric Vehicle Charging ◽

Advanced Students ◽

Doctoral Dissertations

This book discusses many modern, cutting-edge applications of graph theory, such as traffic networks and Braess’ paradox, navigable networks and optimal routing for emergency response, backbone/dominating sets in wireless sensor networks, placement of electric vehicle charging stations, pedestrian safety and graph-theoretic methods in molecular epidemiology. Because of the rapid growth of research in this field, the focus of the book is on the up-to-date development of the aforementioned applications. The book will be ideal for researchers, engineers, transport planners and emergency response specialists who are interested in the recent development of graph theory applications. Moreover, this book can be used as teaching material for postgraduate students because, in addition to up-to-date descriptions of the applications, it includes exercises and their solutions. Some of the exercises mimic practical, real-life situations. Advanced students in graph theory, computer science or molecular epidemiology may use the problems and research methods presented in this book to develop their final-year projects, master’s theses or doctoral dissertations; however, to use the information effectively, special knowledge of graph theory would be required.

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