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DAG‐Width and Circumference of Digraphs
Author(s) -
BangJensen Jørgen,
Larsen Tilde My
Publication year - 2016
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21894
Subject(s) - digraph , combinatorics , mathematics , circumference , disjoint sets , bounded function , conjecture , linkage (software) , discrete mathematics , class (philosophy) , computer science , mathematical analysis , geometry , biochemistry , chemistry , artificial intelligence , gene
We prove that every digraph of circumference l has DAG‐width at most l . This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014). 1 As a consequence of this result we deduce that the k ‐linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang‐Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k ‐linkage problem (where we ask for arc‐disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP‐hard on digraphs of DAG‐width at most 5.