Related papers: Art gallery problem with rook and queen vision
A polygonal art gallery can be observed by guards placed at one third of its corners. However, the strategy of placing guards at every third corner does not work for all art galleries. In this note, we provide an example of a nine-sided art…
We prove that every simply connected orthogonal polygon of $n$ vertices can be partitioned into $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of…
We analyze Solo Chess puzzles, where the input is an $n \times n$ board containing some standard Chess pieces of the same color, and the goal is to make a sequence of capture moves to reduce down to a single piece. Prior work analyzes this…
We resolve the complexity of the point-boundary variant of the art gallery problem, showing that it is $\exists\mathbb{R}$-complete, meaning that it is equivalent under polynomial time reductions to deciding whether a system of polynomial…
A partial Latin square of order $n$ can be represented by a $3$-dimensional chess-board of size $n\times n\times n$ with at most $n^2$ non-attacking rooks. Based on this representation, we apply a uniform method to prove the M. Hall's,…
The author presents two tricks to accelerate depth-first search algorithms for a class of combinatorial puzzle problems, such as tiling a tray by a fixed set of polyominoes. The first trick is to implement each assumption of the search with…
If n is a multiple of 4, then a square of side n can be tiled with T-tetrominos, using a well-known construction. If n is even but not a multiple of four, then there exists an equally well-known construction for tiling a square of side n…
In the chapter "Magic with a Matrix" in \emph{Hexaflexagons and Other Mathematical Diversions} (1988), Martin Gardner describes a delightful "party trick" to fill the squares of a $d$-by-$d$ chessboard with nonnegative integers such that…
In this paper we study queen's graphs, which encode the moves by a queen on an $n\times m$ chess board, through the lens of chip-firing games. We prove that their gonality is equal to $nm$ minus the independence number of the graph, and…
It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right…
We consider the number of domino tilings of an odd-by-odd rectangle that leave one hole. This problem is equivalent to the number of near-perfect matchings of the odd-by-odd rectangular grid. For any particular position of the vacancy on…
There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see an entire rectangle, or along a staircase, or along an orthogonal path with at most $k$ bends. In this paper, we study all these…
We introduce the notion of a normal gallery, a gallery in which any configuration of guards that visually covers the walls covers the entire gallery. We show that any star gallery is normal and any gallery with at most two reflex corners is…
Victor Klee introduce the art gallery problem during a conference in Stanford in August 1976 with that question: "How many guards are required to guard an art gallery?" In 1987, Ghosh provided an approximation algorithm for vertex guards…
We study the properties of the rook complex $\mathcal{R}$ of a polyomino $\mathcal{P}$ seen as independence complex of a graph $G$, and the associated Stanley--Reisner ideal $I_\mathcal{R}$. In particular, we characterize the polyominoes…
The art gallery problem enquires about the least number of guards sufficient to ensure that an art gallery, represented by a simple polygon $P$, is fully guarded. Most standard versions of this problem are known to be NP-hard. In 1987,…
The n-queens puzzle is a well-known combinatorial problem that requires to place n queens on an n x n chessboard so that no two queens can attack each other. Since the 19th century, this problem was studied by many mathematicians and…
Polyominoes are a subset of polygons which can be constructed from integer-length squares fused at their edges. A system of polygons P is interlocked if no subset of the polygons in P can be removed arbitrarily far away from the rest. It is…
In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph $G$. All players occupy vertices of $G$. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does…
We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and say that a polyomino is crystallized if…