is a nonempty graph $P$ whose vertices can be numbered so that $$V(P) = \{x_0,x_1,...,x_k\},\ k \geqslant 1, \quad \text{and} \quad E(P) = \{\{x_0,x_1\},...,\{x_{k-1},x_k\}\}.$$ In this case we say that $P$ is a path joining $x_0$ and $x_k$.

A graph $G$ is *finite* if $V(G)$ is a finite set, $|V(G)| < \infty$. In this paper we consider finite graphs only. A finite graph $C$ is a *cycle* if $|V(C)|\geq 3$ and there exists an enumeration $v_1$, $v_2$, $\ldots$, $v_n$ of its vertices such that $$(\{v_i,v_j\}\in E(C))\Leftrightarrow (|i-j|=1\quad \mbox{or}\quad |i-j|=n-1).$$ A graph $H$ is a *subgraph* of a graph $G$ if $$V(H) \subseteq V(G) \quad \text{and} \quad E(H) \subseteq E(G).$$ We write $H\subseteq G$ if $H$ is a subgraph of $G$.

A graph $G$ is *connected* if for every two distinct $u$, $v \in V(G)$ there is a path $P \subseteq G$ joining $u$ and $v$. A connected graph without cycles is called a *tree*. A vertex $v$ of a tree $T$ is a *leaf* if the *degree* of $v$ is less than two, $$\delta(v) = |\{u \in V(T) \colon \{u, v\} \in E(T)\}| < 2.$$ If a vertex $v$ is not a leaf of $T$, then we say that $v$ 