On the diameter of dot-critical graphs

A graph G is $k$-dot-critical (totaly $k$-dot-critical) if $G$ is dot-critical (totaly dot-critical) and the domination number is $k$. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a $k$-dot-critical graph and a totally $k$-dot-critical graph $G$ with no critical vertices for $k \geq 4$? We find the best bound for the diameter of a $k$-dot-critical graph, where $k \in\{4,5,6\}$ and we give a family of $k$-dot-critical graphs (with no critical vertices) with sharp diameter $2k-3$ for even $k \geq 4$

Total Domination Dot Critical and Dot Stable Graphs.

Two vertices are said to be identifed if they are combined to form one vertex whose neighborhood is the union of their neighborhoods. A graph is total domination dot-critical if identifying any pair of adjacent vertices decreases the total domination number. On the other hand, a graph is total domination dot-stable if identifying any pair of adjacent vertices leaves the total domination number unchanged. Identifying any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most two. Among other results, we characterize total domination dot-critical trees with total domination number three and all total domination dot-stable graphs

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Domination changing and unchanging signed graphs upon the vertex removal

A subset S of V (Σ) is a dominating set of Σ if |N⁺(v) ∩ S| > |N⁻(v) ∩ S| for all v ∈ V − S. This article is to start a study of those signed graphs that are stable and critical in the following way: If the removal of an arbitrary vertex does not change the domination number, the signed graph will be stable. The signed graph, on the other hand, is unstable if an arbitrary vertex is removed and the domination number changes. Specifically, we analyze the change in the domination of the vertex deletion and stable signed graphs.Publisher's Versio

DETERMINATION OF THE RESTRAINED DOMINATION NUMBER ON VERTEX AMALGAMATION AND EDGE AMALGAMATION OF THE PATH GRAPH WITH THE SAME ORDER

Graph theory is a mathematics section that studies discrete objects. One of the concepts studied in graph theory is the restrained dominating set which aims to find the restrained dominating number. This research was conducted by examining the graph operation result of the vertex and edges amalgamation of the path graph in the same order. The method used in this research is the deductive method by using existing theorems to produce new theorems that will be proven deductively true. This research aimed to determine the restrained dominating number in vertex and edges amalgamation of the path graph in the same order. The results obtained from this study are in the form of the theorem about the restrained dominating number of vertex and edges amalgamation of the path graph in the same order, namely: for , ⌋, and for , ⌋

Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z \geq N^{-1+\epsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices

Placing Monitoring Devices in Electric Power Networks Modeled by Block Graphs.

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graph theory. A set S of vertices is defined to be a power dominating set of a graphs if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. In this thesis, we investigate the power domination number of a block graph