Induced Subgraphs of Johnson Graphs

TL;DR

This paper explores the properties and characterizations of induced subgraphs of Johnson graphs, providing conditions, algorithms, and relationships with other graph classes, and establishing foundational results in this new area of graph theory.

Contribution

It introduces the concept of induced subgraphs of Johnson graphs (JIS), provides necessary and sufficient conditions for a graph to be JIS, and develops an algorithm to identify them.

Findings

All trees, cycles, and complete graphs are JIS.

Disjoint unions and Cartesian products of JIS are JIS.

Every JIS graph of order n is an induced subgraph of J(m,2n) for some m <= n.

Abstract

The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS for short) do not seem to have been studied before. We give some necessary conditions and some sufficient conditions for a graph to be JIS, including: in a JIS graph, any two maximal cliques share at most two vertices; all trees, cycles, and complete graphs are JIS; disjoint unions and Cartesian products of JIS graphs are JIS; every JIS graph of order n is an induced subgraph of J(m,2n) for some m <= n. This last result gives an algorithm for deciding if a graph is JIS. We also show that all JIS graphs are edge move distance graphs, but not vice versa.