A Peterson Graph Problem in C

In this article, we will discuss a Peterson Graph Problem in C with its algorithm and implementation.

Introduction:

It takes an organized strategy to tackle the complicated nature associated with this special mathematical construct, whereas resolving a Peterson graph problem with a C program. Having 10 vertices and 15 edges, the Peterson structure is an intriguing illustration of an incredibly tiny yet extremely symmetric network named following the Danish mathematician Julius Petersen. Its unique structure and characteristics make it an appealing challenge in computing terms.

In this graph, first the requirement is imperative that the program in C build a representation of the Peterson graph. Using an adjacency matrix or adjacency list represents a single such technique. An adjacency matrix is commonly selected for such a brief graph to maximize performance and simplicity. The graph's vertex connections can be readily preserved and manipulated due to this matrix.

Initializing the graph representation is the following procedure at once it has been defined. During the initialization procedure, the computer program inputs the correct values onto the adjacency matrix to appropriately symbolize the edges of the Peterson graph. The groundwork for ensuing graph manipulations and analysis are established by this setup.

The foundation of every network-related issue solving endeavor is the graph traversal and search algorithms. Algorithms including Depth-First Search (DFS) and Breadth-First Search (BFS) can potentially be used, contingent upon the particular issue that has to be solved on the Peterson graph. These techniques make it straightforward to explore the foundation of the graph, spot trends, and solve varied graph-related problems.

In addition, the C program has the ability to incorporate logic, particular regarding the Peterson graph. This reasoning is essential for accomplishing the intended results, independent of the goal, detecting cycles, figuring out connections, or addressing additional graph-related issues. The application may effectively make use of the special properties of the Peterson graph to address particular concerns by implementing problem-specific deductive reasoning into practice.

These components come together in the C program's main function because it runs and tests the developed features. The software exhibits its ability to solve Peterson graph issues accurately and efficiently through careful development and implementation. As an outcome, a C program may effectively handle a range of difficulties represented by the Peterson graph by applying an organized methodology and making use of basic graph theory ideas.

Algorithm:

In graph theory, the Peterson graph has become renowned for its intriguing characteristics. Understanding if a given graph is isomorphic to the Peterson graph that is a common challenge related to it. Since there currently is no known polynomial-time mixture to tackle the problem at hand, it is said to be NP-complete.

function isIsomorphic(graph1, graph2):
    for each vertex v in graph1:
        for each vertex u in graph1:
            if degree(v) != degree(u) in graph1:
                return false
            if degree(v) != degree(u) in graph2:
                return false

    for each vertex v in graph1:
        for each vertex u in graph1:
            if v is adjacent to u in graph1:
                if corresponding vertices in graph2 are not adjacent:
                    return false
            if v is not adjacent to u in graph1:
                if corresponding vertices in graph2 are adjacent:
                    return false

    return true

petersonGraph = [[0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
                 [1, 0, 1, 0, 0, 0, 1, 0, 0, 0],
                 [0, 1, 0, 1, 0, 0, 0, 1, 0, 0],
                 [0, 0, 1, 0, 1, 0, 0, 0, 1, 0],
                 [1, 0, 0, 1, 0, 0, 0, 0, 0, 1],
                 [1, 0, 0, 0, 0, 0, 0, 1, 1, 0],
                 [0, 1, 0, 0, 0, 0, 0, 0, 1, 1],
                 [0, 0, 1, 0, 0, 1, 0, 0, 0, 1],
                 [0, 0, 0, 1, 0, 1, 1, 0, 0, 0],
                 [0, 0, 0, 0, 1, 0, 1, 1, 0, 0]]

graph2 = [[0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
          [1, 0, 1, 0, 0, 0, 1, 0, 0, 0],
          [0, 1, 0, 1, 0, 0, 0, 1, 0, 0],
          [0, 0, 1, 0, 1, 0, 0, 0, 1, 0],
          [1, 0, 0, 1, 0, 0, 0, 0, 0, 1],
          [1, 0, 0, 0, 0, 0, 0, 1, 1, 0],
          [0, 1, 0, 0, 0, 0, 0, 0, 1, 1],
          [0, 0, 1, 0, 0, 1, 0, 0, 0, 1],
          [0, 0, 0, 1, 0, 1, 1, 0, 0, 0],
          [0, 0, 0, 0, 1, 0, 1, 1, 0, 0]]

if isIsomorphic(petersonGraph, graph2):
    print("The second graph is isomorphic to the Peterson graph.")
else:
    print("The second graph is not isomorphic to the Peterson graph.")

Example:

Let us take an example to illustrate the Peterson Graph Problem in C.

#include <stdio.h>
#define V 10 // Number of vertices in the Peterson graph
// Function to check if two graphs are isomorphic
int isIsomorphic(int graph1[V][V], int graph2[V][V]) {
    for (int i = 0; i < V; i++) {
        for (int j = 0; j < V; j++) {
            if (graph1[i][j] != graph2[i][j]) {
               return 0; // Not isomorphic
            }
        }
    }
    return 1; // Isomorphic
}

int main() {
    // Define the Peterson graph adjacency matrix
    int petersonGraph[V][V] = {
        {0, 1, 0, 0, 1, 1, 0, 0, 0, 0},
        {1, 0, 1, 0, 0, 0, 1, 0, 0, 0},
        {0, 1, 0, 1, 0, 0, 0, 1, 0, 0},
        {0, 0, 1, 0, 1, 0, 0, 0, 1, 0},
        {1, 0, 0, 1, 0, 0, 0, 0, 0, 1},
        {1, 0, 0, 0, 0, 0, 0, 1, 1, 0},
        {0, 1, 0, 0, 0, 0, 0, 0, 1, 1},
        {0, 0, 1, 0, 0, 1, 0, 0, 0, 1},
        {0, 0, 0, 1, 0, 1, 1, 0, 0, 0},
        {0, 0, 0, 0, 1, 0, 1, 1, 0, 0}
    };

    // Define another graph to check for isomorphism (a copy of Peterson graph)
    int graph2[V][V] = {
        {0, 1, 0, 0, 1, 1, 0, 0, 0, 0},
        {1, 0, 1, 0, 0, 0, 1, 0, 0, 0},
        {0, 1, 0, 1, 0, 0, 0, 1, 0, 0},
        {0, 0, 1, 0, 1, 0, 0, 0, 1, 0},
        {1, 0, 0, 1, 0, 0, 0, 0, 0, 1},
        {1, 0, 0, 0, 0, 0, 0, 1, 1, 0},
        {0, 1, 0, 0, 0, 0, 0, 0, 1, 1},
        {0, 0, 1, 0, 0, 1, 0, 0, 0, 1},
        {0, 0, 0, 1, 0, 1, 1, 0, 0, 0},
        {0, 0, 0, 0, 1, 0, 1, 1, 0, 0}
    };

    // Check if the second graph is isomorphic to the Peterson graph
    if (isIsomorphic(petersonGraph, graph2)) {
        printf("The second graph is isomorphic to the Peterson graph.\n");
    } else {
        printf("The second graph is not isomorphic to the Peterson graph.\n");
    }

    return 0;
}

Output:

The second graph is isomorphic to the Peterson graph.

Explanation:

In this example, we ascertain that two graphs are isomorphic and implemented in the C code. In this example, the first thing to do is to define a constant V, which is set to 10 for the Peterson graph. This constant V represents the total number of vertices in the graphs. The matrices for adjacency of two topologies are compared using the isIsomorphic function. It goes over each matrix's component sequentially and determines if they are equal. The method returns 0 whenever any matching components are different, meaning that the graphs are not isomorphic. If not, it yields 1, which denotes the isomorphism of the graphs. Two different versions of the Peterson graph have been defined as the matrices of adjacency in the main function.

While the second one (graph2) is a duplicate of the Peterson graph, the first one (petersonGraph) is supplied explicitly. The next step in the program is to call the isIsomorphic function and pass in the Peterson graph and graph2 to see if they are isomorphic. Lastly, dependent on the function's return result, it produces a message indicating if the second graph is isomorphic to the Peterson graph. The "The second graph is isomorphic to the Peterson graph." is printed if the function is called and returns a value of 1; else, "The second graph doesn't appear to be isomorphic to the Peterson graph." is printed.

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