3 problems

Let $G$ be a plane graph, and let $P$ be a set of non-crossing single-touch shortest paths in $G$. The path covering with forests number of $P$, denoted by $\operatorname{PCFN}(P)$…

Let $t_{k-1}(n)$ denote the maximum number of edges in a $K_{k-1}$-free graph on $n$ vertices, and let $\tau_k(G)$ be the minimum size of a vertex set meeting every copy of $K_k$ i…

Let $G$ be a graph on $n$ vertices. Write $\tau_3(G)$ for the minimum size of a vertex set meeting every copy of $K_3$ in $G$. Let $s>t\geq 1$ be fixed integers, and let…