Abstract
We investigate the complexity of finding a minimum Steiner tree in new subclasses of split graphs namely tree-convex split graphs and circular-convex split graphs. It is known that the Steiner tree problem (STREE) is NP-complete on split graphs [1]. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique (K), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set (I). Further, we show that STREE is polynomial-time solvable for path (triad)-convex split graphs with convexity on I, and circular-convex split graphs. Finally, we show that STREE can be used as a framework for the dominating set problem in split graphs, and hence the complexity of STREE and the dominating set problem is the same for all these graph classes.
This work is partially supported by the DST-ECRA Project-ECR/2017/001442.
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Mohanapriya, A., Renjith, P., Sadagopan, N. (2022). P Versus NPC: Minimum Steiner Trees in Convex Split Graphs. In: Balachandran, N., Inkulu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2022. Lecture Notes in Computer Science(), vol 13179. Springer, Cham. https://doi.org/10.1007/978-3-030-95018-7_10
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