IJPAM: Volume 29, No. 3 (2006)

TOTAL SIGNED DOMINATION NUMBERS OF GRAPHS

Xinzhong Lu
Department of Mathematics
Zhejiang Normal University
Jinhua, 321004, P.R. CHINA
e-mail: lvxingzhong@163.com


Abstract.Let $G$ be a finite connected simple graph with vertex set $V(G)$ and edge set $E(G)$. A total signed domination function of $G$ is a function $f$ : $V(G)\cup E(G) \rightarrow \{-1,1\}$. The weight of $f$ is $w(f)=\sum_{x\in V(G)\cup E(G)}f(x)$. For an element $x\in V(G)\cup E(G)$, we define $f[x]=\sum_{y\in N_T[x]}f(y)$. A total signed domination function of $G$ is a function $f$ : $V(G)\cup E(G) \rightarrow \{-1,1\}$ such that $f[x]\geq 1$ for all $x\in V(G)\cup E(G)$. The total signed domination number $\gamma_s^*(G)$ of $G$ is the minimum weight of a total signed domination function on $G$.

In this paper, we obtained some lower bounds for the total signed domination number of a graph $G$ and computed exact values of $\gamma_s^*(G)$ when $G$ are $C_n$ and $P_n$.

Received: April 21, 2006

AMS Subject Classification: 26A33

Key Words and Phrases: total signed domination function, total signed domination number

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 29
Issue: 3