Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
An optimal algorithm for finding a maximum independent set of a circular-arc graph
SIAM Journal on Computing
Stability in circular arc graphs
Journal of Algorithms
Achromatic number is NP-complete for cographs and interval graphs
Information Processing Letters
Linear time algorithms on circular-arc graphs
Information Processing Letters
Precoloring extension. I: Interval graphs
Discrete Mathematics - Special volume (part 1) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs”
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
An O(n2 algorithm for circular-arc graph recognition
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
A Polynomial Solution to the Undirected Two Paths Problem
Journal of the ACM (JACM)
Wiring Knock-Knee Layouts: A Global Approach
IEEE Transactions on Computers
A New Linear Algorithm for the Two Path Problem on Chordal Graphs
Proceedings of the Eighth Conference on Foundations of Software Technology and Theoretical Computer Science
On universally easy classes for NP-complete problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Note: on universally easy classes for NP-complete problems
Theoretical Computer Science
Algorithms for interval structures with applications
FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management
Induced disjoint paths in AT-Free graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Induced disjoint paths in claw-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Algorithms for interval structures with applications
Theoretical Computer Science
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Arikati, Pandu Rangan, and Manacher (BIT, 31 (1991) 182-193) developed an O(m+n) algorithm to find two vertex disjoint paths in a circular arc graph on n vertices and m edges. We provide an improved solution to this problem: the algorithm presented here is both faster (O(n) time complexity) and simpler than the previous algorithm. The method involves reductions to interval graphs. In an interval graph, the critical notions are unordered paths (vertex disjoint paths from s1, s2 to t1, t2 in either order) and interchangeable paths (existence of both pairs of vertex disjoint paths). We also prove a theorem (which is best possible, in a sense), that guarantees existence of vertex disjoint paths, if arcs are sufficiently dense. Finally, we show that the more general problem of determining the existence of k vertex disjoint paths (from s1,.., sk to t1,..., tk) is NP-complete, where k is part of the input, even restricted co the class of interval graphs.