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In this paper we initiate the study of total restrained domination in graphs. Let $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by ${\gamma }_r^t\left(G\right)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for ${\gamma }_r^t\left(G\right)$ are given in Section 2. Then the Nordhaus-Gaddum-type...

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ${\left({\phi }_p\left(u^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\right)=0$, $0<t<1$, where ${\phi }_p\left(s\right):={\left|s\right|}^{p-2}s$, $p>1$, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient $q\left(t\right)$ may be singular at $t=0,1$.

The independent domination number $i\left(G\right)$ (independent number $\beta \left(G\right)$) is the minimum (maximum) cardinality among all maximal independent sets of $G$. Haviland (1995) conjectured that any connected regular graph $G$ of order $n$ and degree $\delta \le \frac{1}{2}n$ satisfies $i\left(G\right)\le \lceil \frac{2n}{3\delta }\rceil \frac{1}{2}\delta$. For $1\le k\le l\le m$, the subset graph $S_m(k,l)$ is the bipartite graph whose vertices are the $k$- and $l$-subsets of an $m$ element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for $i\left(S_m(k,l)\right)$ and prove that...