Wiener index and Steiner 3-Wiener index of a graph

Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the Steiner $k$-Wiener index, hence for $k=2$ we get the Wiener index. The modular graphs are graphs in which every three vertices $x, y$ and $z$ have at least one median vertex $m(x,y,z)$ that belongs to shortest paths between each pair of $x, y$ and $z$. The Steiner 3-Wiener index of a modular graph is expressed in terms of its Wiener index. As a corollary formulae for the Steiner 3-Wiener index of Fibonacci and Lucas cubes are obtained

m_3^3-Convex geometries are A-free

Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V, then the elements of M are called convex sets and the pair (V, M) is called an aligned space. If S is a subset of V, then the convex hull of S is the smallest convex set that contains S. Suppose X in M. Then x in X is an extreme point for X if X-x is in M. The collection of all extreme points of X is denoted by ex(X). A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G=(V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)-U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. A set S of vertices in a graph is m_k-convex if it contains the monophonic interval of every k-set of vertices is S. A set of vertices S of a graph is m^3-convex if for every pair u,v of vertices in S, the vertices on every induced path of length at least 3 are contained in S. A set S is m_3^3-convex if it is both m_3- and m^3- convex. We show that if the m_3^3-convex sets form a convex geometry, then G is A-free.Comment: 15 pages, 4 figure

Steiner Distance in Graphs--A Survey

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. In this paper, we summarize the known results on the Steiner distance parameters, including Steiner distance, Steiner diameter, Steiner center, Steiner median, Steiner interval, Steiner distance hereditary graph, Steiner distance stable graph, average Steiner distance, and Steiner Wiener index. It also contains some conjectures and open problems for further studies.Comment: 85 pages, 14 figures, 3 table