Let \( G \) be a connected graph and let \( U \) be a set of vertices in \( G \). A \emph{minimal \( U \)-tree} is a subtree \( T \) of \( G \) that contains \( U \) and has the property that every vertex of \( V(T) – U \) is a cut-vertex of \( \langle V(T) \rangle \). The \emph{monophonic interval} of \( U \) is the collection of all vertices of \( G \) that lie on some minimal \( U \)-tree. A set \( S \) of vertices of \( G \) is \( m_k \)-\emph{convex} if it contains the monophonic interval of every \( k \)-subset \( U \) of vertices of \( S \). Thus \( S \) is \( m_2 \)-convex if and only if it is \( m \)-convex.

In this paper, we consider three local convexity properties with respect to \( m_3 \)-convexity and characterize the graphs having either property.

Let \( G \) and \( H \) be graphs on \( n+2 \) vertices \( \{u_1, u_2, \ldots, u_n, x, y\} \) such that \( G – u_i \cong H – u_i \), for \( i = 1, 2, \ldots, n \). Recently, Ramachandran, Monikandan, and Balakumar have shown in a sequence of two papers that if \( n \geq 9 \), then \( |\varepsilon(H) – \varepsilon(G)| \leq 1 \). In this paper, we present a simpler proof of their theorem, using a counting lemma.