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Graph Homomorphism
Introduction
Graph homomorphism may be a crucial concept in chart hypothesis and computational science. Within the setting of C dialect, a chart homomorphism may be a mapping between two charts that jam the contiguousness connections between their vertices. It is frequently spoken to as a work that allocates vertices from one chart to vertices in another, whereas keeping up the edges between them. This concept empowers the consideration and examination of basic similitudes and associations between distinctive charts. By executing chart homomorphism in C, software engineers can investigate different applications, such as chart coordinating, chart coloring, and chart isomorphism testing, contributing to the field of chart hypothesis and related computational errands.
Approach 1: Backtracking Algorithm
Step 1 − Initialize a purge mapping cluster.
Step 2 − Begin with the primary vertex of the primary chart.
Step 3 − For each vertex within the moment chart, check in case it can be mapped to the current vertex.
Step 4 − In case a substantial mapping is found, include it in the mapping array and recursively continue to the following vertex.
Step 5 − In the event that a mapping isn't conceivable, backtrack and attempt another conceivable mapping.
Step 6 − Rehash steps 3-5 until all vertices are mapped or no substantial mapping is found.
Example
#include <stdbool.h>
#include <stdio.h>
#define MAX_VERTICES 100
// Function to check if a mapping preserves adjacency relationships
bool isHomomorphism(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int mapping[], int n, int v, int u) {
for (int i = 0; i < v; i++) {
if (graph1[v][i] && graph2[u][mapping[i]] == 0) {
return false;
}
}
return true;
}
// Backtracking approach to find graph homomorphism
bool findHomomorphismUtil(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int mapping[], int n, int v) {
if (v == n) {
return true; // All vertices mapped
}
for (int u = 0; u < n; u++) {
if (isHomomorphism(graph1, graph2, mapping, n, v, u)) {
mapping[v] = u; // Add mapping
if (findHomomorphismUtil(graph1, graph2, mapping, n, v + 1)) {
return true;
}
mapping[v] = -1; // Remove mapping (backtrack)
}
}
return false;
}
bool findHomomorphism(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int n) {
int mapping[MAX_VERTICES];
// Initialize mapping with -1
for (int i = 0; i < n; i++) {
mapping[i] = -1;
}
return findHomomorphismUtil(graph1, graph2, mapping, n, 0);
}
int main() {
// Example graphs (adjacency matrices)
int graph1[MAX_VERTICES][MAX_VERTICES] = {
{0, 1, 0},
{1, 0, 1},
{0, 1, 0}
};
int graph2[MAX_VERTICES][MAX_VERTICES] = {
{0, 1, 1},
{1, 0, 0},
{1, 0, 0}
};
int n = 3; // Number of vertices
if (findHomomorphism(graph1, graph2, n)) {
printf("A graph homomorphism exists.\n");
} else {
printf("No graph homomorphism exists.\n");
}
return 0;
}
Output
A graph homomorphism exists.
Approach 2: Constraint Satisfaction Problem (CSP) Algorithm
Step 1 − Make a space for each vertex within the moment chart, at first containing all vertices of the primary chart.
Step 2 − Emphasize each vertex within the, to begin with, graph and allot esteem from the space of the comparing vertex within the moment chart, while fulfilling contiguousness imperatives.
Step 3 − In the event that all vertices are doled out values, return genuine. Something else, backtrack and attempt the following conceivable task.
Step 4 − Rehash steps 2-3 until a substantial task is found or all conceivable outcomes are depleted.
Example
#include <stdbool.h>
#include <stdio.h>
#include <string.h>
#define MAX_VERTICES 100
// Function to check if a mapping preserves adjacency relationships
bool isHomomorphism(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int mapping[], int n, int v, int u) {
for (int i = 0; i < v; i++) {
if (graph1[v][i] && graph2[u][mapping[i]] == 0) {
return false;
}
}
return true;
}
// CSP-based approach to find graph homomorphism
bool findHomomorphismUtil(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int mapping[], bool domain[][MAX_VERTICES], int n, int v) {
if (v == n) {
return true; // All vertices assigned values
}
for (int u = 0; u < n; u++) {
if (domain[v][u] && isHomomorphism(graph1, graph2, mapping, n, v, u)) {
mapping[v] = u; // Assign value
bool newDomain[n][MAX_VERTICES];
memcpy(newDomain, domain, sizeof(bool) * n * MAX_VERTICES);
// Update domain for adjacent vertices
for (int i = 0; i < n; i++) {
if (graph1[v][i]) {
for (int j = 0; j < n; j++) {
newDomain[i][j] = newDomain[i][j] && (j == u);
}
}
}
if (findHomomorphismUtil(graph1, graph2, mapping, newDomain, n, v + 1)) {
return true;
}
}
}
return false;
}
bool findHomomorphism(int graph1[MAX_VERTICES][MAX_VERTICES], int graph2[MAX_VERTICES][MAX_VERTICES], int n) {
int mapping[MAX_VERTICES];
bool domain[MAX_VERTICES][MAX_VERTICES];
// Initialize mapping with -1 and domain with true
for (int i = 0; i < n; i++) {
mapping[i] = -1;
for (int j = 0; j < n; j++) {
domain[i][j] = true;
}
}
return findHomomorphismUtil(graph1, graph2, mapping, domain, n, 0);
}
int main() {
// Example graphs (adjacency matrices)
int graph1[MAX_VERTICES][MAX_VERTICES] = {
{0, 1, 0},
{1, 0, 1},
{0, 1, 0}
};
int graph2[MAX_VERTICES][MAX_VERTICES] = {
{0, 1, 1},
{1, 0, 0},
{1, 0, 0}
};
int n = 3; // Number of vertices
if (findHomomorphism(graph1, graph2, n)) {
printf("A graph homomorphism exists.\n");
} else {
printf("No graph homomorphism exists.\n");
}
return 0;
}
Output
No graph homomorphism exists.
Conclusion
In conclusion, Chart Homomorphism may be a noteworthy concept in graph theory and computational arithmetic. Executing it within the C dialect permits the investigation of basic similitudes and associations between distinctive charts. We examined two approaches for finding chart homomorphism: backtracking, and Constraint Satisfaction Problem (CSP). These calculations empower the distinguishing proof of mappings that protect contiguousness connections between vertices. Encourage optimization or elective calculations are worth considering for real-world applications.
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