## Induced subgraphs of Johnson graphs.(English)Zbl 1246.05048

Summary: The Johnson graph $$J(n,N)$$ is defined as the graph whose vertices are the $$n$$-subsets of the set $$\{1,2,\dots,N\}$$, 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 $$m \leq n$$. 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.

### MSC:

 05C12 Distance in graphs 05C62 Graph representations (geometric and intersection representations, etc.) 05C85 Graph algorithms (graph-theoretic aspects)
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