Abstract
The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following 
(shortly, 
) problem. The input is a graph \(G=(V,E)\), with each vertex in the set V having an assigned color, “
” or “
”. We seek a maximum-cardinality subset \(V'\subseteq V\) of vertices that is 
(having exactly \(|V'|/2\) red nodes and \(|V'|/2\) blue nodes), such that the subgraph induced by the vertex set \(V'\) in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2-coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.
S. Bhore—Partially supported by the Lynn and William Frankel Center for Computer Science, Ben-Gurion University of the Negev, Israel.
J. S. B. Mitchell—Support from the National Science Foundation (CCF-1526406) and the US-Israel Binational Science Foundation (project 2016116).
S. Pandit—Partially supported by the Indo-US Science & Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with grant number 2017/94, Department of Science and Technology, Government of India.
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Acknowledgement
We thank Florian Sikora for pointing out the connection with the Graph Motif problem.
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Bhore, S., Chakraborty, S., Jana, S., Mitchell, J.S.B., Pandit, S., Roy, S. (2019). The Balanced Connected Subgraph Problem. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_17
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