Notes on dual-critical graphs
Abstract
We define dual-critical graphs as graphs having an acyclic orientation, where the indegrees are odd except for the unique source. We have very limited knowledge about the complexity of dual-criticality testing. By the definition the problem is in NP, and a result of Balázs and Christian Szegedy provides a randomized polynomial algorithm, which relies on formal matrix rank computing. It is unknown whether dual-criticality test can be done in deterministic polynomial time. Moreover, the question of being in co-NP is also open. We give equivalent descriptions for dual-critical graphs in the general case, and further equivalent descriptions in the special cases of planar graphs and 3-regular graphs. These descriptions provide polynomial algorithms for these special classes. We also give an FPT algorithm for a relaxed version of dual-criticality called $k$-dual-criticality.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2014
- DOI:
- arXiv:
- arXiv:1410.1653
- Bibcode:
- 2014arXiv1410.1653K
- Keywords:
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- Computer Science - Data Structures and Algorithms;
- Computer Science - Computational Complexity;
- Mathematics - Combinatorics
- E-Print:
- 10 pages, conference