Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in $[0,1/2].

Presenting GECO : an eyetracking corpus of monolingual and bilingual sentence reading

This paper introduces GECO, the Ghent Eye-tracking Corpus, a monolingual and bilingual corpus of eye-tracking data of participants reading a complete novel. English monolinguals and Dutch-English bilinguals read an entire novel, which was presented in paragraphs on the screen. The bilinguals read half of the novel in their first language, and the other half in their second language. In this paper we describe the distributions and descriptive statistics of the most important reading time measures for the two groups of participants. This large eye-tracking corpus is perfectly suited for both exploratory purposes as well as more directed hypothesis testing, and it can guide the formulation of ideas and theories about naturalistic reading processes in a meaningful context. Most importantly, this corpus has the potential to evaluate the generalizability of monolingual and bilingual language theories and models to reading of long texts and narratives

Throttling for the game of Cops and Robbers on graphs

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $O(\sqrt n)$.Comment: 22 pages, 4 figure