The Total Interval Number of a Graph II: Trees and Complexity

A multiple-interval representation of a simple graph G assigns each vertex a union of disjoint real intervals, such that vertices are adjacent if and only if their assigned sets intersect. The total interval number I (G) is the minimum of the total number of intervals used in any such representation of G. For triangle-free graphs, I (G) = m + t(G), where m is the number of edges in G and t(G) is the minimum number of pairwise edge-disjoint trails such that every edge of G has an endpoint in at least one of the trails. This yields the NP-completeness of testing I (G) = m + 1, even for triangle-free 3-regular planar graphs, and an alternative proof that HAMILTONIAN CYCLE is NP-complete for line graphs. It also yields a linear-time algorithm to compute I (G) for trees and a characterization of the trees requiring m + t intervals, for fixed t. Further corollaries include the Aigner/Andreae bound of I (G) ≤ (5n − 3)/4 for n-vertex trees (achieved by subdividing every edge of a star), a characterization of the extremal trees, and a shorter proof of the extremal bound (5m + 2)/4 for connected graphs. Ke ywords: intersection graphs, intervals, trees, extremal problem AMS Subject Classification: 05C05, 05C35, 05C85 Running head: TOTAL INTERVAL NUMBER, II 1Research supported in part by ONR Grant N00014-85K0570.