The Four-Distance Domination Number in the Ladder and Star Graphs Amalgamation Result and Applications

The study purpose is to determine the four-distance domination number in the amalgamation operation graph, namely the vertex amalgamation result graph of ladder graph Amal(L_m,v,n) with m≥2 and n≥2 and the vertex amalgamation result graph of a star graph with its name Amal(S_m,v,n) with m≥2 and n>2. In addition, the application use the Four-distance domination number on Jember Regency Covid-19 taskforce post-placement. The Importanceof this research, namely the optimal distribution of the Covid-19 task force post. It is not just doing mask surgeries every day on the streets. The optimal referred to can be in the form of integrated handlers in each sub-district or points that are considered to need fast handling so that coordination between posts can respond and immediately identify cases of transmission and potential infections due to interactions with patients who are already positive. The methods used in this research are pattern recognition and axiomatic deductive methods. The results of this study include:γ_4 (Amal(S_m,v,n))=1; for m≥2 and n≥2, γ_4 (Amal(L_m,v,n))={■(1; for 2≤m≤4 @⌊m/8⌋n+1 for m≡0,1,2,3,4 (mod 8)@⌈m/8⌉n; for others m ) ┤ and based on the Indonesia Country, Jember Regency Map, 2 Covid 19 task-force posts are needed to be placed in Balung and Kalisat sub-districts using the Four-distance domination number application.

The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable

Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets of any such family of sets are interesting from a combinatorial point of view and have algorithmic applications. Many such bounds on different families of sets over different classes of graphs are already provided in the literature. In particular, Rote recently showed that the number of minimal dominating sets in trees of order $n$ is at most $95^{\frac{n}{13}}$ and that this bound is asymptotically sharp up to a multiplicative constant. We build on his work to show that what he did for minimal dominating sets can be done for any family of sets definable by a monadic second order formula. We first show that, for any monadic second order formula over graphs that characterizes a given kind of subset of its vertices, the maximal number of such sets in a tree can be expressed as the \textit{growth rate of a bilinear system}. This mostly relies on well known links between monadic second order logic over trees and tree automata and basic tree automata manipulations. Then we show that this "growth rate" of a bilinear system can be approximated from above.We then use our implementation of this result to provide bounds on the number of independent dominating sets, total perfect dominating sets, induced matchings, maximal induced matchings, minimal perfect dominating sets, perfect codes and maximal irredundant sets on trees. We also solve a question from D. Y. Kang et al. regarding $r$-matchings and improve a bound from G\'orska and Skupie\'n on the number of maximal matchings on trees. Remark that this approach is easily generalizable to graphs of bounded tree width or clique width (or any similar class of graphs where tree automata are meaningful)

Growth of Replacements

The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some 2 balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some n, the object of the game is to find the maximum possible sum of values g(n) for a table of n balls. In this article, the sequence g(n)/n for n=1,2,… will be shown to converge to a growth rate λ. Furthermore, this value λ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate λ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given λ0 is smaller than λ is provided to approximate λ. When the values of the balls are all rational, we can compute the precise value of λ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit λ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements)

Growth of Bilinear Maps

We study a problem that is algebraic in nature but has certain applications in graph theory. It can be seen as a generalization of the joint spectral radius. Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, both with nonnegative coefficients and entries, among an exponential number of ways to combine $n$ instances of $s$ using $n-1$ applications of $*$, we are interested in the largest possible entry in a resulting vector. Let $g(n)$ denote this value, the asymptotic behaviour of $g(n)$ is investigated through the growth rate $$\lambda=\limsup_{n\to\infty} \sqrt[n]{g(n)}.$$ It is known that checking $\lambda\le 1$ is undecidable, as a consequence of the corresponding fact for the joint spectral radius. However, efficient algorithms are available to compute it exactly in certain cases, or approximate it to any precision in general. Furthermore, when the vector $s$ is positive, there exists some $r$ so that \const n^{-r}\lambda^n\le g(n)\le \const n^r\lambda^n. It means $\lambda$ is actually a limit when $s>0$. However, checking if this is the case in general is also undecidable. Some types of patterns for optimal combinations are proposed and studied as well, with some connections to the finiteness property of a set of matrices. The techniques that are used for our problem can be applied well for the joint spectral radius, and they produce some stronger results by even simpler arguments. For example, if $\|\Sigma^n\|$ denotes the largest possible entry in a product of $n$ matrices drawn from a finite set $\Sigma$ of nonnegative matrices, whose joint spectral radius is denoted by $\rho(\Sigma)$, then there exists some $r$ so that \[ \const n^r\rho(\Sigma)^n\le \|\Sigma^n\|\le \const n^r\rho(\Sigma)^n. \

γ\gamma-Graphs of Trees

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in $G(\gamma)$ if they differ by a single vertex and the two different vertices are adjacent in $G$. In this paper, we consider $\gamma$-graphs of trees. We develop an algorithm for determining the $\gamma$-graph of a tree, characterize which trees are $\gamma$-graphs of trees, and further comment on the structure of $\gamma$-graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.Comment: 22 pages, 3 figure