Abstract
A graph \(G\,=\,(V,E)\) is called almost bipartite if \(G\) is not bipartite, but there exists a vertex \(v\in V\) such that \(G-\{v\}\) is bipartite. We consider the problem of testing if \(G\) is almost bipartite or not.
This problem arises from the study on the \(k\)-arch layout problem. It is known that, given a graph \(G\) and an integer \(k\ge 2\), it is NP-complete to determine if \(G\) has a \(k\)-arch layout. On the other hand, \(G\) has a 1-arch layout if and only if \(G\) is almost bipartite [3]. It is straightforward to test if \(G\) is almost bipartite in \(O(n(n+m))\) time by using depth first search.
In this paper, we present a simple linear time algorithm for solving this problem. The efficiency of the algorithm is achieved by sophisticated applications of depth first search tree and the study of the structure of such graphs.
Research supported in part by NSF Grant CCR-1319732.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boundy, J.A., Murty, U.S.R.: Graph Theory with Applications. Elsevier, North Holland (1976)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
Dujmović, M., Wood, D.R.: On linear layout of graphs. Discrete Math. Theoret. Comput. Sci. 6, 339–358 (2004)
Iwata, Y., Oka, K., Yoshida, Y.: Linear-time FPT algorithms via network flow. In: Proceedings of the SODA, vol. 2014, pp. 1749–1761 (2014)
Proömel, H.J., Schickinger, T., Steger, A.: A note on triangle-free and bipartite graphs. Discrete Math. 257(2–3), 531–540 (2002)
Ramanujan, M.S., Saurabh, S.: Linear time parameterized algorithms via skew-symmetric multicuts. In: Proceedings of the SODA, vol. 2014, pp. 1739–1748 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
He, D., He, X. (2015). A Linear Time Algorithm for Determining Almost Bipartite Graphs. In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-17142-5_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17141-8
Online ISBN: 978-3-319-17142-5
eBook Packages: Computer ScienceComputer Science (R0)