On the unit interval number of a graph

Thomas Andreae
1988 Discrete Applied Mathematics  
The unit Interval number of a simple unduected graph G, denoted r,(G), IS the least nonnegatrve integer r for which we can assign to each vertex m G a collectIon of at most r closed umt intervals on the real hne such that two distinct vertices u and w of G are adjacent If and only If some Interval for u intersects some interval for w. This concept generahzes the notion of a unit interval graph m the same way as the previously studted Interval number generahzes the notion of an interval graph We
more » ... present the followmg results on the umt interval number Let G be a graph on n vertices Then r,(G) s r+(n -I)1 For even n, the extremal graphs are K" n_ 1 and C4 For odd II zz 3, the extremal graphs are Cs , those graphs which contam induced copies of KI" _ 2 and (If n = 5) those graphs with an induced C4. These results suggest the questlon whether the unit interval number 1s unbounded for claw-free graphs, which we answer m the affirmative On the other hand, we fmd that l,(G)5 3 when G 1s the complement of a forest. In addltlon, we also present an upper bound on r,(G) m terms of the edge number of G, together with a charactenzation of the correspondmg extremal graphs
doi:10.1016/0166-218x(88)90118-7 fatcat:5bwazx6oxjaupjf2w2qamjdytq