### فهرست مطالب

• Volume:9 Issue: 1, 2020
• تاریخ انتشار: 1398/12/11
• تعداد عناوین: 5
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• Pouyeh Sharifani, Mohammad Reza Hooshmandasl * Pages 1-24

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.

*The formula is not displayed correctly!
Keywords: domination, Total domination, [1, Total [1, 2]-set, Series-parallel graphs, Generalized series-parallel graph
• Siamak Yassemi * Pages 25-30
‎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.
Keywords: simplicial complex‎, ‎Leray simplicial complex‎, ‎Regularity
• Kinkar Chandra Das, Mostafa Tavakoli * Pages 31-39

‎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.

* The formula in not displayed correctly!

Keywords: Deleted lexicographic product, Metric dimension, Defensive k-alliance