Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.A Linear Algorithm for Computing $\gamma_{_{[1,2]}}$-set in Generalized Series-Parallel Graphs
http://toc.ui.ac.ir/article_24185_4442.html
For a graph $G=(V,E)$, a set $S subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v in V setminus S$ is dominated by at most two vertices of $S$, i.e. $1 leq vert N(v) cap S vert leq 2$. Moreover a set $S subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 leq vert N(v) cap S vert leq 2$. The $[1,2]$-domination number of $G$, denoted $gamma_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $gamma_{[1,2]}(G)$ is called a $gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $gamma_{[1,2]}$-set and a $gamma_{t[1,2]}$-set in generalized series-parallel graphs.Sat, 29 Feb 2020 20:30:00 +0100On a generalization of Leray simplicial complexes
http://toc.ui.ac.ir/article_24257_4442.html
‎We define a refinement of the notion of Leray simplicial complexes and study its properties‎. ‎Moreover‎, ‎we translate some of our results to the language of commutative algebra‎.Sat, 29 Feb 2020 20:30:00 +0100Bounds for metric dimension and defensive $k$-alliance of graphs under deleted lexicographic product
http://toc.ui.ac.ir/article_24081_4442.html
‎Metric dimension and defensive $k$-alliance number are two distance-based graph invariants‎ ‎which have applications in robot navigation‎, ‎quantitative analysis of secondary RNA structures‎, ‎national defense and fault-tolerant computing‎. ‎In this paper‎, ‎some bounds for metric‎ ‎dimension and defensive $k$-alliance of deleted lexicographic product of graphs are presented‎. ‎We also show that the bounds are sharp‎.Sat, 29 Feb 2020 20:30:00 +0100Nilpotent graphs of skew polynomial rings over non-commutative rings
http://toc.ui.ac.ir/article_24321_4442.html
Let $R$ be a ring and $alpha$ be a ring endomorphism of $R$‎. ‎The undirected nilpotent graph of $R$‎, ‎denoted by $Gamma_N(R)$‎, ‎is a graph with vertex set $Z_N(R)^*$‎, ‎and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent‎, ‎where $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ In this article‎, ‎we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;alpha]$ and the graph-theoretical properties of its nilpotent graph $Gamma_N(R[x;alpha])$‎. ‎It is shown that if $R$ is a symmetric and $alpha$-compatible with exactly two minimal primes‎, ‎then $diam(Gamma_N(R[x,alpha]))=2$‎. ‎Also we prove that $Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $Z_2timesZ_2$‎.Sat, 29 Feb 2020 20:30:00 +0100$H$-kernels by walks in subdivision digraph
http://toc.ui.ac.ir/article_24341_0.html
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph)‎. ‎A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$‎. ‎A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks)‎. ‎Suppose that $D$ is a digraph possibly infinite‎. ‎In this paper we will work with the subdivision digraph $S_H$($D$) of $D$‎, ‎where $S_H$($D$) is an $H$-colored digraph defined as follows‎: ‎$V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)}‎, ‎where ($u$‎, ‎$a$‎, ‎$v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$)‎. ‎We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$)‎.Sat, 04 Jan 2020 20:30:00 +0100Connected zero forcing sets and connected propagation time of graphs
http://toc.ui.ac.ir/article_24400_0.html
The zero forcing number $Z(G)$ of a graph $G$is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$to be colored (black) after some times. ``color change rule'': a white vertex is changed to a blackvertex when it is the only white neighbor of a black vertex.In this case, we say that the black vertex forces the white vertex.In this paper, we investigate the concept of connected zero forcing setand connected zero forcing number. We discusses this subject for special graphs and some products of graphs.Also we introduce the connected propagation time. Graphs with extreme minimum connected propagationtimes and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.Sat, 18 Jan 2020 20:30:00 +0100Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$
http://toc.ui.ac.ir/article_24401_4442.html
In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.Sat, 29 Feb 2020 20:30:00 +0100