The distribution of distances in the star graph \( S{T_n} \) (\(1 < n \in \mathbb{Z}\)) is established, and subsequently a threaded binary tree is obtained that realizes an orientation of \( S{T_n} \) whose levels are given by the distances to the identity permutation, via a pruning algorithm followed by a threading algorithm. In the process, the distributions of distances of the efficient dominating sets of \( S{T_n} \) are determined.

A set \(D\) of vertices in a graph \(G = (V, E)\) is a locating-dominating set if for every two vertices \(u, v\) in \(V \setminus D\), the sets \(N(u) \cap D\) and \(N(v) \cap D\) are non-empty and different. We establish two equivalent conditions for trees with unique minimum locating-dominating sets.

Let \( [n]^* \) denote the set of integers \(\{-\frac{n-1}{2}, \ldots, \frac{n-1}{2}\}\) if \(n\) is odd, and \(\{-\frac{n}{2}, \ldots, \frac{n}{2}\} \setminus \{0\}\) if \(n\) is even. A super edge-graceful labeling \(f\) of a graph \(G\) of order \(p\) and size \(q\) is a bijection \(f : E(G) \to [q]^*\), such that the induced vertex labeling \(f^*\) given by \(f^*(u) = \sum_{uv \in E(G)} f(uv)\) is a bijection \(f^* : V(G) \to [p]^*\). A graph is super edge-graceful if it has a super edge-graceful labeling. We prove that total stars and total cycles are super edge-graceful.

A total dominating function (TDF) of a graph \( G = (V, E) \) is a function \( f : V \to [0,1] \) such that for all \( v \in V \), the sum of the function values over the open neighborhood of \( v \) is at least one. A minimal total dominating function (MTDF) \( f \) is a TDF such that \( f \) is not a TDF if the value of \( f(v) \) is decreased for any \( v \in V \). A convex combination of two MTDFs \( f \) and \( g \) of a graph \( G \) is given by \( h_\lambda = \lambda f + (1-\lambda)g \), where \( 0 < \lambda < 1 \). A basic minimal total dominating function (BMTDF) is an MTDF which cannot be expressed as a convex combination of two or more different MTDFs. In this paper, we study the structure of the set of all minimal total dominating functions (\(\mathfrak{F}_T\)) of some classes of graphs and characterize the graphs having \(\mathfrak{F}_T\) isomorphic to one simplex.

Vertex elimination orderings play a central role in many portions of graph theory and are exemplified by the so-called `perfect elimination orderings’ of chordal graphs. But perfect elimination orderings and chordal graphs enjoy many special advantages that overlap in more general settings: the random way that simplicial vertices can be chosen, always having a choice of simplicial vertices, the hereditary nature of being simplicial, and the neutral effect of deleting a simplicial vertex on whether the graph is chordal. A graph metatheory of vertex elimination formalizes such distinctions for general vertex elimination and examines them with simple theorems and delineating counterexamples.