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Linear Time Algorithms for Happy Vertex Coloring Problems for Trees

  • N. R. Aravind
  • Subrahmanyam Kalyanasundaram
  • Anjeneya Swami KareEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

Given an undirected graph \(G = (V, E)\) with \(|V| = n\) and a vertex coloring, a vertex v is happy if v and all its neighbors have the same color. An edge is happy if its end vertices have the same color. Given a partial coloring of the vertices of the graph using k colors, the Maximum Happy Vertices (also called k-MHV) problem asks to color the remaining vertices such that the number of happy vertices is maximized. The Maximum Happy Edges (also called k-MHE) problem asks to color the remaining vertices such that the number of happy edges is maximized. For arbitrary graphs, k-MHV and k-MHE are NP-Hard for \(k \ge 3\). In this paper we study these problems for trees. For a fixed k we present linear time algorithms for both the problems. In general, for any k the proposed algorithms take \(O(nk \log k)\) and O(nk) time respectively.

Keywords

Happy vertex Happy edge Graph coloring Coloring trees 

Notes

Acknowledgement

We thank the anonymous reviewers for their detailed reviews and suggestions.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • N. R. Aravind
    • 1
  • Subrahmanyam Kalyanasundaram
    • 1
  • Anjeneya Swami Kare
    • 1
    Email author
  1. 1.Department of Computer Science and Engineering Indian Institute of TechnologyHyderabadIndia

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