Forcing signed domination numbers in graphs

Abstract

We initiate the study of forcing signed domination in graphs. Afunction $f:V(G)⟶\{-1,+1\}$ is called {ıt signeddominating function} if for each $vınV(G)$, ${ßsize\sum }_{uınN[v]}f(u)\ge 1$. For a signed dominating function $f$ of $G$, the{ıt weight} $f$ is $w(f)={ßsize\sum }_{vınV}f(v)$. The {ıt signeddomination number} ${\gamma }_s(G)$ is the minimum weight of a signeddominating function on $G$. A signed dominating function of weight${\gamma }_s(G)$ is called a ${\gamma }_s(G)$-{ıt function}. A${\gamma }_s(G)$-function $f$ can also be represented by a set ofordered pairs $S_f=\{(v,f(v)):vınV\}$. A subset $T$ of $S_f$ iscalled a {ıt forcing subset\/} of $S_f$ if $S_f$ is the uniqueextension of $T$ to a ${\gamma }_s(G)$-function. The {ıt forcingsigned domination number} of $S_f$, $f(S_f,{\gamma }_s)$, is definedby $f(S_f,{\gamma }_s)=min\{|T|:T\text{ is a forcing subset of/ }{S}_f\}$ and the {ıt forcing signed domination number} of $G$,$f(G,{\gamma }_s)$, is defined by$f(G,{\gamma }_s)=min\{f(S_f,{\gamma }_s):S_f\text{tx}isa{\gamma }_s(G)\text{-function}\}$. For every graph $G$,$f(G,{\gamma }_s)\ge 0$. In this paper we show that for integer $a,b$with $a$ positive, there exists a simple connected graph $G$ suchthat $f(G,{\gamma }_s)=a$ and ${\gamma }_s(G)=b$. The forcing signeddomination number of several classes of graph, including paths,cycles, Dutch-windmills, wheels, ladders and prisms are determined.