In this paper we initiate a systematic study of a problem that has the flavor of two classical problems, namely \sc Coloring} and {\sc Domination}, from the perspective of algorithms and complexity. A {\it dominator coloring} of a graph G is an assignment of colors to the vertices of G such that it is a proper coloring and every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of G is called the {\it dominator chromatic number} of G and is denoted by χ_d(G). In the {\sc Dominator Coloring (DC)} problem, a graph G and a positive integer k are given as input and the objective is to check whether χ_d(G)≤q k. We first show that unless P=NP, DC cannot be solved in polynomial time on bipartite, planar, or split graphs. This resolves an open problem posed by Chellali and Maffray [{\it Dominator Colorings in Some Classes of Graphs, Graphs and Combinatorics, 2011] about the polynomial time solvability of DC on chordal graphs. We then complement these hardness results by showing that the problem is fixed parameter tractable (FPT) on chordal graphs and in graphs which exclude a fixed apex graph as a minor

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In recent years, several Electronic Design Automation (EDA)

All-solutions ATPG' based methods huve found applications in Model Checking sequential Circuits, and they can also improve the defect coverage of a test-suite, by generating distinct multiple-detect patterns. Conventional decision selection heuristics and leaming techniques for an ATPG engine were originally developed to 'quickly 'find any available (single) solution. Such decision selection heuristics may not be the best for an 'all-solutions ATPG' engine, where all the solutions need to be found. In this paper, we explore new techniques to guide an 'all-solutions ATPG engine'. We first present a new decision selection heuristic that makes use ofthe 'connectivity of gates' in the circuit in order to obtain a compact solution-set. Next, we analyze the 'symmetry in search-states' that was exploited in 'SuccessDriven Leaming' [ I ] and extend it to prune conflict subspaces as well. Finally, we propose a new metric that determines the use of leamt information a priori. This information is stored and used efliciently during 'success driven leaming '. Experimental results show that we can compute the complete solution-set with our new heuristics for large ISCAS '89 and ITC '99 circuits, where conventional guidance heuristics fail.