University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 On the defensive alliances in graph 1 14 23227 10.22108/toc.2018.50156.1396 EN Hasan Kharazi Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran. Alireza Mosleh Tehrani Department of Mathematics, Faculty of Science, Iran University of Science and Technology, Tehran, Iran. Journal Article 2016 03 17 ‎Let \$ G = (V,E) \$ be a graph‎. ‎We say that \$ S subseteq V \$ is a defensive alliance if for every \$ u in S \$‎, ‎the number of neighbors \$ u \$ has in \$ S \$ plus one (counting \$ u \$) is at least as large as the number of neighbors it has outside \$ S \$‎. ‎Then‎, ‎for every vertex \$ u \$ in a defensive alliance \$ S \$‎, ‎any attack on a single vertex by the neighbors of \$ u \$ in \$ V-S \$ can be thwarted by the neighbors of \$ u \$ in \$ S \$ and \$ u \$ itself‎. ‎In this paper‎, ‎we study alliances that are containing a given vertex \$ u \$ and study their mathematical properties‎. http://toc.ui.ac.ir/article_23227_49f13d333028b939ed2d096a98282fcf.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey 15 40 23166 10.22108/toc.2018.112665.1585 EN Majid Arezoomand University of Larestan Alireza Abdollahi University of Isfahan Pablo Spiga Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Journal Article 2018 08 25 Fixed-point-free permutations‎, ‎also known as derangements‎, ‎have been studied for centuries‎. ‎In particular‎, ‎depending on their applications‎, ‎derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics‎: ‎from number theory to algebraic graph theory‎. ‎Substantial progress has been made on the study of derangements‎, ‎many long-standing open problems have been solved‎, ‎and many new research problems have arisen‎. ‎The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs‎. ‎The methods used in this area range from deep group theory‎, ‎including the classification of the finite simple groups‎, ‎to combinatorial techniques‎. ‎This article is devoted to surveying results‎, ‎open problems and methods in this area‎. http://toc.ui.ac.ir/article_23166_1e1c1fe183cadcd86904ba2543084f1f.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 On the zero forcing number of generalized Sierpinski graphs 41 50 23265 10.22108/toc.2018.101107.1463 EN Ebrahim Vatandoost Imam Khomeini International University Fatemeh Ramezani Yazd University Saeid Alikhani Yazd University‎ Journal Article 2016 12 17 In this article we study the Zero forcing number of Generalized Sierpi'{n}ski graphs \$S(G,t)\$‎. ‎More precisely‎, ‎we obtain a general lower bound on the Zero forcing number of \$S(G,t)\$ and we show that this bound is tight‎. ‎In particular‎, ‎we consider the cases in which the base graph \$G\$ is a star‎, ‎path‎, ‎a cycle or a complete graph‎. http://toc.ui.ac.ir/article_23265_e768f70fa89d95c17111a4dc08270b06.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 On the double bondage number of graphs products 51 59 23167 10.22108/toc.2018.114111.1605 EN Hamidreza Maimani Zeinab Koushki Mathematics, research and science, tehran Journal Article 2018 11 19 A set \$D\$ of vertices of graph \$G\$ is called \$double\$ \$dominating\$ \$set\$ if for any vertex \$v\$, \$|N[v]cap D|geq 2\$. The minimum cardinality of \$double\$ \$domination\$ of \$G\$ is denoted by \$gamma_d(G)\$. The minimum number of edges \$E'\$ such that \$gamma_d(Gsetminus E)>gamma_d(G)\$ is called the double bondage number of \$G\$ and is denoted by \$b_d(G)\$. This paper determines that \$b_d(Gvee H)\$ and exact values of \$b(P_ntimes P_2)\$, and generalized corona product of graphs. http://toc.ui.ac.ir/article_23167_c10b003a8aa01309879f4e72bf73d795.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 8 1 2019 03 01 A generalization of global dominating function 61 68 23550 10.22108/toc.2019.110404.1562 EN Mostafa Momeni Department of‎ ‎Mathematics‎, ‎Shahid Rajaee Teacher Training University‎, ‎P.O‎. ‎Box 16785-163, Tehran‎, ‎Iran Ali Zaeembashi Department of math, Shahid Rajaee Teacher Training University, Tehran, Iran Journal Article 2018 04 11 Let \$G\$ be a graph‎. ‎A function \$f‎ : ‎V (G) longrightarrow {0,1}\$‎, ‎satisfying‎ ‎the condition that every vertex \$u\$ with \$f(u) = 0\$ is adjacent with at‎ ‎least one vertex \$v\$ such that \$f(v) = 1\$‎, ‎is called a dominating function \$(DF)\$‎. ‎The weight of \$f\$ is defined as \$wet(f)=Sigma_{v in V(G)} f(v)\$‎. ‎The minimum weight of a dominating function of \$G\$‎ ‎is denoted by‎ ‎\$gamma (G)\$‎, ‎and is called the domination number of \$G\$‎. ‎A dominating‎ ‎function \$f\$ is called a global dominating function \$(GDF)\$ if \$f\$ is‎ ‎also a \$DF\$ of \$overline{G}\$‎. ‎The minimum weight of a global dominating function is denoted by‎ ‎\$gamma_{g}(G)\$ and is called global domination number of \$G\$‎. ‎In this paper we introduce a generalization of global dominating function‎. ‎Suppose \$G\$ is a graph and \$sgeq 2\$ and \$K_n\$ is the complete graph on \$V(G)\$‎. ‎A function \$ f:V(G)longrightarrow { 0,1} \$ on \$G\$ is \$s\$-dominating function \$(s-DF)\$‎, ‎if there exists some factorization \${G_1,ldots,G_s }\$ of \$K_n\$‎, ‎such that \$G_1=G\$ and \$f\$ is dominating function of each \$G_i\$‎. http://toc.ui.ac.ir/article_23550_0c12856c79206ad93e86a934875ed0e3.pdf