Optimization Problems in Dotted Interval Graphs

Abstract

The class of D-dotted interval (D-DI) graphs is the class of intersection graphs of arithmetic progressions with jump (common difference) at most D. We consider various classical graph-theoretic optimization problems when these are restricted to D-DI graphs of arbitrarily fixed D. We show that Maximum Independent Set, Minimum Vertex Cover, and Minimum Dominating Set can be solved in polynomial time in this graph class, answering an open question posed by Jiang [17]. We also show that Minimum Vertex Cover can be approximated within a factor of (1 + ε) for any ε > 0 in linear time. This algorithm generalizes to a wide class of deletion problems including the classical Minimum Feedback Vertex Set and Minimum Planar Deletion problems. Our algorithms are based on classical results in algorithmic graph theory and new structural properties of D-DI graphs that may be of independent interest.

Characterization of 2-Path Signed Network

“…e intersection graphs or networks [1,2] form a large family of structures which include many important network such as interval [3,4], permutation [5,6], chordal [7,8], circular-arc [9,10], circle [11], string [12], line [13,14], and path [15,16]. Most of these networks are of great significance not only theoretically but because of their applicability in the fields such as transportation [17], wireless networking [18], scheduling problem [19], molecular biology [20], circuit routing [21], and sociology.…”