This paper considers the existence of short synchronizing words in deterministic finite automata (DFAs). We define two general strategies for generating synchronizing words, and we show that each of these strategies can be applied if and only if a DFA is synchronizable. Furthermore, we show that if a synchronizable DFA is well-structured, then our strategies generate short synchronizing words. The first of our strategies, called the cornering strategy , takes advantage of states in a DFA with properties similar to those of a polytope vertex. The second of our strategies, similar to the cornering strategy and called the f-ordered strategy , takes advantage of a partial order defined on the states of a DFA. We apply our cornering strategy to the class of difference DFAs , whose states form subsets of R d and whose input symbols correspond to translation vectors between states. We show that difference DFAs share many similarities with aperiodic DFAs, and in particular, a difference DFA M has a synchronizing word if and only if it has a universally reachable state. Using the cornering strategy, we also show that under certain conditions, such an n -state DFA M has a synchronizing word of length at most ( n − 1 ) 2 and thereby satisfies Černý’s conjecture. Using the f -ordered strategy, we also show that a synchronizable DFA whose states have a certain partial order that is preserved by a set of short words also has a short synchronizing word, and we consider several consequences of this result. Finally, we consider how the cornering strategy can be applied to the problem of synchronizing the product of two DFAs M 1 , M 2 that share a common alphabet, and we show that the product M 1 × M 2 often has a synchronizing word that is subquadratic in the number of states of M 1 × M 2 .

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph G , the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by cc ( G ) . We give a sufficient condition for a triangle-free graph to have attacking cop number at most 2 and we characterise when outerplanar graphs have attacking cop number 2. We also prove that all bipartite planar graphs G have cc ( G ) ≤ 4 and show this is tight by constructing a bipartite planar graph G with cc ( G ) = 4 . Finally we construct 17 non-isomorphic graphs H of order 58 with cc ( H ) = 6 and c ( H ) = 3 . This provides the first example of a graph H with cc ( H ) − c ( H ) ≥ 3 , extending work by Bonato et al. (2014). We conclude with a list of conjectures and open problems.