In this paper we study the GRAPH ISOMORPHISM problem on graphs of bounded treewidth, bounded degree, or bounded bandwidth. GRAPH ISOMORPHISM can be solved in polynomial time for graphs of bounded treewidth, pathwidth, or bandwidth, but the exponent depends on the treewidth, pathwidth, or bandwidth. Thus, we look for special cases where "fixed parameter tractable" polynomial time algorithms can be established. We introduce some new and natural graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve GRAPH ISOMORPHISM in O(n 2 ) time for graphs with bounded rooted path distance width, and in O(n 3 ) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is NP-hard, but both path and tree distance width can be computed in O(n k+1 ) time, when they are bounded by a constant k; the rooted path or tree distance width can be computed in O(ne) time. Finally, we study the relationships between the newly introduced parameters and other existing graph parameters.
The classes A r and S r are de ned as the classes of those graphs, where the minimum degree greedy algorithm always approximates the maximum independent set MIS problem within a factor of r, respectively, where this algorithm has a sequence of choices that yield an output that is at most a factor r from optimal, r 1 a rational number. It is shown that deciding whether a given graph belongs to A r is coNP-complete for any xed r 1, and deciding whether a given graph belongs to S 1 is DP-hard, and belongs to 2 P. Also, the MIS problem remains NP-complete when restricted to S r .
We explore the feasibility of early introduction to automata theory through gamification. We designed a puzzle game that players can answer correctly if they understand the fundamental concepts of automata theory. In our investigation, 90 children played the game, and their actions were recorded in play logs. An analysis of the play logs shows that approximately 60% of the children achieved correct-answer rates of at least 70%, which suggests that primary and lower secondary school students can understand the fundamental concepts of automata theory. Meanwhile, our analysis shows that most of them do not
fully
understand automata theory, but some of them have a good understanding of the concept.
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