Related papers: A note on watchman's walks in de Bruijn graphs
Given two non-adjacent vertices \( u \) and \( v \), we say a $uv$-walk \( W \) dominates a $uv$-walk \( W' \) if every internal vertex of \( W' \) is adjacent to some internal vertex of \( W \) or belongs to \( W \). A class of walks…
This article rigorously analyzes the meeting time between pursuers and evaders performing random walks on digraphs. There exist several bounds on the expected meeting time between random walkers on graphs in the literature, however,…
We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera.…
We consider the watchman route problem for multiple watchmen in staircase polygons, which are rectilinear $x$- and $y$-monotone polygons. For two watchmen, we propose an algorithm to find an optimal solution that takes quadratic time,…
We consider the problem of inferring an edge-labeled graph from the sequence of edge labels seen in a walk of that graph. It has been known that this problem is solvable in $O(n \log n)$ time when the targets are path or cycle graphs. This…
We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the…
The well-known \textsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the \textsc{$k$-Watchmen…
We analyze continuous-time quantum walks on necklace graphs - cyclical graphs consisting of many copies of a smaller graph (pearl). Using a Bloch-type ansatz for the eigenfunctions, we block-diagonalize the Hamiltonian, reducing the…
The de Bruijn graph $G_K$ of a set of strings $S$ is a key data structure in genome assembly that represents overlaps between all the $K$-length substrings of $S$. Construction and navigation of the graph is a space and time bottleneck in…
The well-known $k$-disjoint path problem ($k$-DPP) asks for pairwise vertex-disjoint paths between $k$ specified pairs of vertices $(s_i, t_i)$ in a given graph, if they exist. The decision version of the shortest $k$-DPP asks for the…
We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. It is…
In the recent research of data mining, frequent structures in a sequence of graphs have been studied intensively, and one of the main concern is changing structures along a sequence of graphs that can capture dynamic properties of data. On…
We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also…
A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a {\it dominating closed neighborhood sequence} if $\{v_1,\, \dots , \,v_k\}$ is a dominating set of $G$ and $N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]$ for every $i$.…
In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in…
We study the behaviour of a sequence of biased random walks X(i), i>=0 on a sequence of random graphs, where the initial graph is Zd and otherwise the graph for the i-th walk is the trace of the (i - 1)-st walk. The sequence of bias vectors…
We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and…
The three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…
Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety…
We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating…