A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood. One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism , answering a question of Chastand, Laviolette and Polat. Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of ‘locally constructible’ (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.

Today's students want to solve problems and experience engineering regardless of where they are - in lecture, in the laboratory, or the dorm room. Professors want to provide a hands-on learning experience to empower students who want to tinker, experiment, and explore concepts while improving the comprehension through reinforcement. Student access to affordable, low-cost technology enables educators to address limitations in the laboratory, including access to equipment, time on task, and cost. With a portable laboratory, a student can learn concepts in their preferred environments and provides a supplement to the traditional lecture and laboratory based courses.

In this paper, various fast transmission strategies for sending information from a source s over a large distance to a target t in ad hoc wireless networks where the nodes are distributed as a Poisson process of intensity is presented. The existence of an infinite component, i.e., percolation, is not sufficient for our problem since the proportion of vertices in the infinite component may be very low. To achieve connectivity the power must increase with the number of vertices, since there is some positive chance that a vertex is isolated. Result shows that with directional transmissions, even with very low power there exist points at arbitrarily large distance that can communicate.