Graph Interval Research Articles

On Strong Intervals in Fuzzy Graphs

Intervals and convexity play crucial roles in the applications of graph theory such as town planning and design of graphics. In this article, the concept of geodetic interval in graphs is extended to fuzzy graphs. Intervals are useful in the study of properties of fuzzy graphs which depend on the geodetic distance between vertices. The axiomatic definition of intervals in fuzzy graphs are used to define intervals in different fuzzy graph structures like fuzzy trees and complete fuzzy graphs. Finally a set theoretic operations of intervals like union, intersection are also discussed and some results are obtained.

GRAPHS WITH 4-STEINER CONVEX BALLS

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set $W$ on $k$ vertices in a connected graph $G$ is a tree with the smallest number of edges in $G$ that contains all vertices of $W$. The Steiner interval $I(W)$ of $W$ consists of all vertices in $G$ that lie on some Steiner tree with respect to $W$. Moreover, a set $S$ of vertices in a graph $G$ is $k$-Steiner convex, denoted $g_k$-convex, if the Steiner interval $I(W)$ of every set $W$ on $k$ vertices is contained in $S$. In this paper we consider two types of local convexities. In particular, for every $k > 3$, we characterize graphs with $g_k$-convex closed neighborhoods around all vertices of the graph. Then we follow with a characterization of graphs with $g_4$-convex closed neighborhoods around all $g_4$-convex sets of the graph.

On certain extensions of intervals in graphs

On certain extensions of intervals in graphs

Intervals in matroid basis graphs

An interval in a graph is a subgraph induced by all the vertices on shortest paths between two given vertices. Intervals in matroid basis graphs satisfy many nice properties. Key results are: (1) any two vertices of a basis graph are together in some longest interval; (2) every basis graph with the minimum number of vertices for its diameter is an interval, indeed a hypercube. (1) turns out to be a simple case of a theorem in Edmonds' theory of matroid partition.