Unique intersectability of diamond-free graphs
2011, Discrete Applied Mathematics
https://doi.org/10.1016/J.DAM.2011.01.012Abstract
For a graph G with vertices v 1 , v 2 ,. .. , v n , a simple set representation of G is a family F = {S 1 , S 2 ,. .. , S n } of distinct nonempty sets such that |S i ∩ S j | = 1 if v i v j is an edge in G, and |S i ∩ S j | = 0 otherwise. Let S(F) = n i=1 S i , and let ω s (G) denote the minimum |S(F)| of a simple set representation F of G. If, for every two minimum simple set representations F and F ′ of G, F can be obtained from F ′ by a bijective mapping from S(F ′) to S(F), then G is said to be s-uniquely intersectable. In this paper, we are concerned with the s-unique intersectability of diamond-free graphs, where a diamond is a K 4 with one edge deleted. Moreover, for a diamond-free graph G, we also derive a formula for computing ω s (G).
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