“DUALLY EXCEPTIONAL” FAMILIES: SPECIFICS, NEEDS AND PROBLEMS

Families blessed with a child with developmental disabilities or a gifted child are not typical families. Such families are exposed to specifics in day-to-day function, establishing interpersonal relationships and fulfilling their family roles. The object of research and studies carried out so far are twice-exceptional individuals, thus excluding the families with one gifted child and another with developmental disabilities. Dually exceptional families have dual challenges in providing additional individualized support for children, in two completely different ways, in two different directions. Based on the results and the analysis of case study of two dually different families, areas in which additional support is required by such families and parents are identified according to family functions, as well as recommendations as to how to empower these areas.

Spanning Trails in a 2-Connected Graph

In this article we prove the following: Let $G$ be a $2$-connected graph with circumference $c(G)$. If $c(G)\leq 5$, then $G$ has a spanning trail starting from any vertex, if $c(G)\leq 7$, then $G$ has a spanning trail. As applications of this result, we obtain the following. Every $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex with the exception of six graphs. Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\emptyset$ and $|S|\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $|S|\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$. Every $2$-connected claw-free graph $G$ with order $n$ and minimum degree $\delta(G)> \frac{n}{7}+4\geq 23$ is traceable or belongs to two exceptional families of well-defined graphs, and moreover, if $\delta(G)> \frac{n}{6}+4\geq 13$, then $G$ is traceable. All above results are sharp in a sense.