Triangular Grid Graphs Research Articles

The equitable non-split domination number of graphs

Finding a group of dominant servers is a necessary step towards optimising service provisioning to all clients in the complex world of server-client networks. This crucial decision rests on the non-split dominance number in the graph architecture. We present the idea of an equitable non-split domination number in graphs to address the possible problem of unequal task distribution among servers. When the vertices of a dominating set, Dens, have the same colour in an equitable colouring scheme, the set is said to be equitable. The induced subgraph ⟨V −D⟩ also maintains its connectivity. In particular, the application of equitable non-split domination numbers to graphs like the Triangular Grid, Antiprism, Wheel, and Complete Graphs is explored in this study.

Resistance values under transformations in regular triangular grids

In Evans and Francis (2022) and Hendel (2021) the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced s times provided the original grid has at least 4s rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously.

A Novel Global Routing Algorithm for Printed Circuit Boards Based on Triangular Grid

Global routing plays a crucial role in printed circuit board (PCB) design and affects the cost of the design significantly. Conventional methods based on rectangular grids have some limitations, whereas this paper introduces a new algorithm that employs a triangular grid model, which offers a more efficient solution to the problem. Firstly, we present a technique to sort all unconnected two-pin nets. Next, a triangular grid graph is constructed to represent the routing resources on the printed circuit board. Finally, we use the concept of maximum flow to identify the paths for global routing and apply detailed routing for the completion of wires. Results from experiments demonstrate that our algorithm is faster than two state-of-the-art routers and does not have any design rule violations for all industrial PCB instances.

TOTAL EDGE IRREGULAR LABELING FOR TRIANGULAR GRID GRAPHS AND RELATED GRAPHS

Let be a graph with and are the set of its vertices and edges, respectively. Total edge irregular -labeling on is a map from to satisfies for any two distinct edges have distinct weights. The minimum for which the satisfies the labeling is spoken as its strength of total edge irregular labeling, represented by . In this paper, we discuss the tes of triangular grid graphs, its spanning subgraphs, and Sierpiński gasket graphs.

Restrained domination and its variants in extended supergrid graphs

Domination problem is a well-known NP-complete problem for general graphs. In this paper, we will study its three variants, including restrained, independent restrained, and restrained-step dominations. A restrained dominating set of a graph is a set R of vertices of the graph such that every vertex of the graph is in R or adjacent to at least a vertex of R, and the subgraph induced by vertices not in R contains no isolated vertex. An independent restrained dominating set of a graph is a restrained dominating set such that its any two vertices are not adjacent. A restrained-step dominating set of a graph is a restrained dominating set such that the subgraph induced by vertices not in this set contains a perfect matching. Restrained domination, independent restrained domination, and restrained-step domination problems are to find a restrained dominating set, independent restrained dominating set, and restrained-step dominating set with the least number of vertices on a graph, respectively. In 2015, we define supergrid graphs for computing the tracks of computer embroidery sewing machine and 3D printer. In this paper, we expand it to the class of extended supergrid graphs. Extended supergrid graphs are a natural extension of supergrid graphs, and they contain grid, triangular grid, and supergrid graphs as subclasses. The complexities of the restrained related domination problems on extended supergrid graphs remain unknown. In this paper, we will prove the restrained and independent restrained domination problems on supergrid graphs is NP-complete. Using the same technique, the restrained domination problem on grid graphs can also be shown to be NP-complete. We then solve the restrained related domination problems on rectangular supergrid graphs, which form a subclass of supergrid graphs, in linear time. Finally, we provide a tight upper bound on the minimum size of (independent) restrained dominating set for rectangular triangular-supergrid graphs forming a subclass of triangular grid graphs.

Two-Step Colouring of Grid Graphs of Different Types

In this article, we consider the NP-hard problem of the two-step colouring of a graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. The optimum two-step colouring is one that uses the minimum possible number of colours.The two-step colouring problem is studied in application to grid graphs. We consider four types of grids: triangular, square, hexagonal, and octogonal. We show that the optimum two-step colouring of hexagonal and octogonal grid graphs requires 4 colours in the general case. We formulate the polynomial algorithms for such a colouring. A square grid graph with the maximum vertex degree equal to 3 requires 4 or 5 colours for a two-step colouring. In the paper, we suggest the backtracking algorithm for this case. Also, we present the algorithm, which works in linear time relative to the number of vertices, for the two-step colouring in 7 colours of a triangular grid graph and show that this colouring is always correct. If the maximum vertex degree equals 6, the solution is optimum.

M-Bonacci graceful labeling

We introduce new labeling called m-bonacci graceful labeling. A graph G on n edges is m-bonacci graceful if the vertices can be labeled with distinct integers from the set such that the derived edge labels are the first n m-bonacci numbers. We show that complete graphs, complete bipartite graphs, gear graphs, triangular grid graphs, and wheel graphs are not m-bonacci graceful. Almost all trees are m-bonacci graceful. We give m-bonacci graceful labeling to cycles, friendship graphs, polygonal snake graphs, and double polygonal snake graphs.

On Triangular Secure Domination Number

Let T_m=(V(T_m), E(T_m)) be a triangular grid graph of m ϵ N level. The order of graph T_m is called a triangular number. A subset T of V(T_m) is a dominating set of T_m if for all u_V(T_m)\T, there exists vϵT such that uv ϵ E(T_m), that is, N[T]=V(T_m). A dominating set T of V(T_m) is a secure dominating set of T_m if for each u ϵ V(T_m)\T, there exists v ϵ T such that uv ϵ E(T_m) and the set (T\{u})ꓴ{v} is a dominating set of T_m. The minimum cardinality of a secure dominating set of T_m, denoted by γ_s(T_m) is called a secure domination number of graph T_m. A secure dominating number γ_s(T_m) of graph T_m is a triangular secure domination number if γ_s(T_m) is a triangular number. In this paper, a combinatorial formula for triangular secure domination number of graph T_m was constructed. Furthermore, the said number was evaluated in relation to perfect numbers.

Robotic Path Planning Based on a Triangular Mesh Map

Aiming at the problem of robot path planning in complex maps, an algorithm of robot path planning based on triangular grid graph is proposed. Firstly, a weighted undirected loop graph and a feasible domain of nodes are obtained by discretizing the triangular mesh map. Next, the Dijkstra search algorithm is applied to find the feasible shortest path from an initial to a final configuration. Finally, The Douglas-Peucker algorithm is applied to remove duplicate and redundant nodes in the feasible path, and the waypoint are extracted. The final path is a curve that is obtained by connecting the several extracted waypoint. The proposed algorithm is tested for various maps. Compared with probabilistic roadmap method, the experimental results show that the proposed method can overcome the shortcomings of the random sampling method. Furthermore, the experimental result of triangular mesh map method used in two labyrinth maps show that the triangular mesh map method can solve the robot path planning problem in complex map very well, and it is an excellent algorithm for robot path planning.

Efficient [formula omitted] broadcast dominating sets of the triangular lattice

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph G called the (t,r)broadcast domination number which depends on the positive integer parameters t and r. In this setting, a v∈V is a broadcast vertex of transmission strengtht if it transmits a signal of strength t−d(u,v) to every vertex u∈V with d(u,v)<t. Given a set of broadcast vertices S⊆V, the reception at vertex u is the sum of the transmissions from the broadcast vertices in S. The set S⊆V is called a (t,r)broadcast dominating set if every vertex u∈V has a reception strength r(u)≥r and for a finite graph G the cardinality of a smallest broadcast dominating set is called the (t,r) broadcast domination number of G. In this paper, we consider the infinite triangular grid graph and define efficient(t,r) broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient (t,r) broadcasts on the infinite triangular grid graph for all t≥r≥1. Using these broadcasts, we then provide upper bounds for the (t,r) broadcast domination numbers for triangular matchstick graphs when (t,r)∈{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)}.

The total H-irregularity strength of triangular ladder and grid graphs

Let graph G admitting cover H. A total k-labeling can be define as total H-irregular k-labeling of the graph G if there is for every two different subgraphs H′ and H” isomorphic to H. We define H-weight as which sum of vertex and edge labels in subgraph H for subgraphs under the total k-labeling . The smallest k value such that G has an H-irregular total k-labeling is the total H-irregularity strength of a graph G, is denoted tHs(G, H). We study the tHs of triangular ladder and grid graphs with H-covering.

On the edge irregularity strength of grid graphs

For a simple graph G, a vertex labeling ϕ:V(G)→{1,2,…,k} is called a vertex k-labeling. For any edge xy in G, its weight wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is called an edge irregular k-labeling of G. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G).In this paper, we determine an exact value of edge irregularity strength for triangular grid graph Lnm, zigzag graph Znm and Cartesian product Pn□Pm□P2.

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.

Domination with decay in triangular matchstick arrangement graphs

We provide results for the exponential dominating numbers and total exponential dominating numbers of a family of triangular grid graphs. We then prove inequalities for these numbers and compare them with inequalities that hold more generally for exponential dominating numbers of graphs.

Hamiltonian cycles in linear-convex supergrid graphs

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. Recently, we have proved the Hamiltonian cycle problem on supergrid graphs to be NP-complete. The Hamiltonian cycle problem on supergrid graphs can be applied to control the stitching trace of computerized sewing machines. In this paper, we will study the Hamiltonian cycle property of linear-convex supergrid graphs which form a subclass of supergrid graphs. A connected graph is called k-connected if there are k vertex-disjoint paths between every pair of vertices, and is called locally connected if the neighbors of each vertex in it form a connected subgraph. In this paper, we first show that any 2-connected, linear-convex supergrid graph is locally connected. We then give constructive proofs to show that any 2-connected, linear-convex supergrid graph contains a Hamiltonian cycle. Based on the constructive proofs, we finally present a linear-time algorithm to construct a Hamiltonian cycle of a 2-connected, linear-convex supergrid graph.

Word-Representability of Face Subdivisions of Triangular Grid Graphs

A graph $$G=(V,E)$$G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if $$(x,y)\in E$$(x,y)źE. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph $$K_4$$K4. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpinski gasket graph is word-representable.

The Hamiltonian properties of supergrid graphs

In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines. In this paper, we will prove that the Hamiltonian cycle problem for supergrid graphs is NP-complete. It is easily derived from the Hamiltonian cycle result that the Hamiltonian path problem on supergrid graphs is also NP-complete. We then show that two subclasses of supergrid graphs, including rectangular (parallelism) and alphabet, always contain Hamiltonian cycles.

Some new characterizations of Hamiltonian cycles in triangular grid graphs

In the studies that have been devoted to the protein folding problem, which is one of the great unsolved problems of science, some specific graphs, like the so-called triangular grid graphs, have been used as a simplified lattice model. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate the thermodynamics of protein folding. In this paper, we present new characterizations of the Hamiltonian cycles in labeled triangular grid graphs, which are graphs constructed from rectangular grids by adding a diagonal to each cell. By using these characterizations and implementing the computational method outlined here, we confirm the existing data, and obtain some new results that have not been published. A new interpretation of Catalan numbers is also included.

Probabilistic Distributed Algorithm for Uniform Election in Triangular Grid Graphs

Probabilistic algorithms are designed to handle problems that do not admit deterministic effective solutions. In the case of the election problem, many algorithms are available and applicable under appropriate assumptions, for example: the uniform election in trees, k??trees and polyominoids. In this paper, first, we introduce a probabilistic algorithm for the uniform election in the triangular grid graphs, then, we expose the set of rules that generate the class of the triangular grid graphs. The main of this paper is devoted to the analysis of our algorithm. We show that our algorithm is totally fair in so far as it gives the same probability to any vertex of the given graph to be elected.

Modeling the spread of fault in majority-based network systems: Dynamic monopolies in triangular grids

In a graph theoretical model of the spread of fault in distributed computing and communication networks, each element in the network is represented by a vertex of a graph where edges connect pairs of communicating elements, and each colored vertex corresponds to a faulty element at discrete time periods. Majority-based systems have been used to model the spread of fault to a certain vertex by checking for faults within a majority of its neighbors. Our focus is on irreversible majority processes wherein a vertex becomes permanently colored in a certain time period if at least half of its neighbors were in the colored state in the previous time period. We study such processes on planar, cylindrical, and toroidal triangular grid graphs. More specifically, we provide bounds for the minimum number of vertices in a dynamic monopoly defined as a set of vertices that, if initially colored, will result in the entire graph becoming colored in a finite number of time periods.