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In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighborhood of a vertex includes itself, and open neighborhoods, where a vertex does not included in its neighborhood. We study the parameterized complexity of conflict-free closed neighborhood coloring and conflict-free open neighborhood coloring problems. We show that both problems are fixed-parameter tractable (FPT) when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Gargano et al. (2015) that conflict-free coloring is fixed-parameter tractable parameterized by the vertex cover number. Also, we show that both problems admit an additive constant approximation algorithm when parameterized by the distance to threshold graphs. We also study the complexity of the problem on special graph classes. We show that both problems can be solved in polynomial time on cographs. For split graphs, we give a polynomial time algorithm for closed neighborhood conflict-free coloring problem, whereas we show that open neighborhood conflict-free coloring is NP-complete. We show that interval graphs can be conflict-free colored using at most four colors.

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Abstract-In this work, we study the correlation between interdisciplinarity of papers within physical sciences and their citations by using meta data of articles published in American Physical Society's Physical Review journals between 1985 to 2012. We use the Weitzman diversity index to measure the diversity of papers and authors, exploiting the hierarchical structure of PACS (Physics and Astronomy Classification Scheme) codes. We find that the fraction of authors with high diversity is increasing with time, where as the fraction of least diversity are decreasing, and moderate diversity authors have higher tendency to switch over to other diversity groups. The diversity index of papers is correlated with the citations they received in a given time period from their publication year. Papers with lower and higher end of diversity index receive lesser citations than the moderate diversity papers.