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Let $G_1$ and $G_2$ be copies of a graph $G$, and let $f:V\left(G_1\right)\to V\left(G_2\right)$ be a function. Then a functigraph $C(G,f)=(V,E)$ is a generalization of a permutation graph, where $V=V\left(G_1\right)\cup V\left(G_2\right)$ and $E=E\left(G_1\right)\cup E\left(G_2\right)\cup \{uv:u\in V\left(G_1\right),v\in V\left(G_2\right),v=f\left(u\right)\}$. In this paper, we study colorability and planarity of functigraphs.

The $P_3$ intersection graph of a graph $G$ has for vertices all the induced paths of order 3 in $G$. Two vertices in $P_3\left(G\right)$ are adjacent if the corresponding paths in $G$ are not disjoint. A $w$-container between two different vertices $u$ and $v$ in a graph $G$ is a set of $w$ internally vertex disjoint paths between $u$ and $v$. The length of a container is the length of the longest path in it. The $w$-wide diameter of $G$ is the minimum number $l$ such that there is a $w$-container of length at most $l$ between any pair of different...