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Let $G$ be a graph with order $p$, size $q$ and component number $\omega$. For each $i$ between $p-\omega$ and $q$, let ${𝒞}_i\left(G\right)$ be the family of spanning $i$-edge subgraphs of $G$ with exactly $\omega$ components. For an integer-valued graphical invariant $\varphi$, if $H\to H^{\text{'}}$ is an adjacent edge transformation (AET) implies $|\varphi \left(H\right)-\varphi \left(H^{\text{'}}\right)|\le 1$, then $\varphi$ is said to be continuous with respect to AET. Similarly define the continuity of $\varphi$ with respect to simple edge transformation (SET). Let $M_j\left(\varphi \right)$ and $m_j\left(\varphi \right)$ be the invariants defined by $M_j\left(\varphi \right)\left(H\right)={max}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$, $m_j\left(\varphi \right)\left(H\right)={min}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$. It is proved that both $M_{p-\omega }\left(\varphi \right)$ and $m_{p-\omega }\left(\varphi \right)$ interpolate...

Let $f$ be an integer-valued function defined on the vertex set $V\left(G\right)$ of a graph $G$. A subset $D$ of $V\left(G\right)$ is an $f$-dominating set if each vertex $x$ outside $D$ is adjacent to at least $f\left(x\right)$ vertices in $D$. The minimum number of vertices in an $f$-dominating set is defined to be the $f$-domination number, denoted by ${\gamma }_f\left(G\right)$. In a similar way one can define the connected and total $f$-domination numbers ${\gamma }_{c,f}\left(G\right)$ and ${\gamma }_{t,f}\left(G\right)$. If $f\left(x\right)=1$ for all vertices $x$, then these are the ordinary domination number, connected domination number and total domination...