arXiv subjects

P. Sharifani

Publications and source records attributed to P. Sharifani.

6 recordsLinked to original sources

arXiv2018

Domination Cover Number of Graphs

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in $O(1.7159^n)$ time. It is known that the number of minimal dominating sets for interval graphs and trees on $n$ vertices is at most $3^{n/3} \approx 1.4422^n$. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set $D$, denoted by $\mathcal{C}_D(G)$, is the summation of the degrees of the vertices in $D$. Maximizing or minimizing this parameter among all minimal dominating sets have interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees, block graphs.

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arXiv2018

An Explicit Construction of Optimal Dominating Sets in Grid

A dominating set in a graph $G$ is a subset of vertices $D$ such that every vertex in $V\setminus D$ is a neighbor of some vertex of $D$. The domination number of $G$ is the minimum size of a dominating set of $G$ and it is denoted by $γ(G)$. Also, a subset $D$ of a graph $G$ is a $[ 1 , 2 ] $-set if, each vertex $v \in V \setminus D$ is adjacent to either one or two vertices in $D$ and the minimum cardinality of $[ 1 , 2 ] $-dominating set of $G$, is denoted by $γ_{[1,2]}(G)$. Chang's conjecture says that for every $16 \leq m \leq n$, $γ(G_{m,n})= \left \lfloor\frac{(n+2)(m+2)}{5}\right \rfloor-4$ and this conjecture has been proven by Goncalves et al. This paper presents an explicit constructing method to find an optimal dominating set for grid graph $G_{m,n}$ where $m,n\geq 16$ in $O(\text{size of answer})$. In addition, we will show that $γ(G_{m,n})=γ_{[1,2]}(G_{m,n})$ where $m,n\geq 16$ holds in response to an open question posed by Chellali et al.

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arXiv2017

Some Results on [1, k]-sets of Lexicographic Products of Graphs

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $γ_{[1, k]}(G)$ is the size of the smallest $[1,k]$-sets of $G$. A set $S'\subseteq V(G)$ is called a total $[1,k]$-set, if for every vertex $v \in V$, $1 \leq | N_G(v) \cap S | \leq k$. If a graph $G$ has at least one total $[1, k]$-set then the cardinality of the smallest such set is denoted by $γ_{t[1, k]}(G)$. We consider $[1, k]$-sets that are also independent. Note that not every graph has an independent $[1, k]$-set. For graphs having an independent $[1, k]$-set, we define $[1, k]$-independence numbers which is denoted by $γ_{i[1, k]}(G)$. In this paper, we investigate the existence of $[1,k]$-sets in lexicographic products $G\circ H$. Furthermore, we completely characterize graphs which their lexicographic product has at least one total $[1,k]$-set. Also, we determine $γ_{[1, k]}(G\circ H)$, $γ_{t[1, k]}(G\circ H)$ and $γ_{i[1, k]}(G\circ H)$. Finally, we show that finding smallest total $[1, k]$-set is $NP$-complete.

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arXiv2017

An Efficient Algorithm for Mixed Domination on Generalized Series-Parallel Graphs

A mixed dominating set $S$ of a graph $G=(V,E)$ is a subset $ S \subseteq V \cup E$ such that each element $v\in (V \cup E) \setminus S$ is adjacent or incident to at least one element in $S$. The mixed domination number $γ_m(G)$ of a graph $G$ is the minimum cardinality among all mixed dominating sets in $G$. The problem of finding $γ_{m}(G)$ is know to be NP-complete. In this paper, we present an explicit polynomial-time algorithm to construct a mixed dominating set of size $γ_{m}(G)$ by a parse tree when $G$ is a generalized series-parallel graph.

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arXiv2017

A Linear Algorithm for Computing $γ_{[1,2]}$-set in Generalized Series-Parallel Graphs

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 $γ_{[1,2]}(G)$,is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $γ_{[1,2]}(G)$ is called a $γ_{[1,2]}$-set. Total $[1,2]$-domination number and $γ_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $γ_{[1,2]}$-set and a $γ_{t[1,2]}$-set in generalized series-parallel graphs.

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arXiv2010

On z-factorization and c-factorization of standard episturmian words

Ziv-Lempel and Crochemore factorization are two kinds of factorizations of words related to text processing. In this paper, we find these factorizations for standard epiesturmian words. Thus the previously known c-factorization of standard Sturmian words is provided as a special case. Moreover, the two factorizations are compared.

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