📘 graph theory

1. **Énoncé du problème** : L'algorithme de Floyd permet de calculer les plus courts chemins entre tous les couples de sommets $(i,j)$ dans un graphe simple sans circuit absorbant,

1. **Stating the problem:** We are asked to analyze the total construction time based on the minimum spanning tree (MST) found previously, considering collision durations and work

1. **Problem statement:** We need to find the minimum number of colors required to color the faces of two nets (A and B) so that no two faces that touch share the same color. 2. **

1. **Problem:** Prove that if $G$ is a tree graph, then the number of vertices $|V(G)|$ equals the number of edges $|E(G)|$ plus 1. 2. **Formula and rules:** A tree is a connected

1. **Problem Statement:** Determine a new network configuration by selecting cables (edges) to retain in the graph connecting districts a, b, c, d, e, f, g, h, i such that all dist

1. **Problem Statement:** We want to construct a CNF Boolean expression $\phi_H$ for graph $H$ such that $\phi_H$ is True if and only if the assignment of truth values to variables

1. **Problem Statement:** We are given two cycles $\eta$ and $\eta'$ in a graph $G$ and asked to understand the Cartesian product graph $\hat{G} = \hat{G}(\eta, \eta')$ formed by t

1. **Problem:** Can five houses be connected to two utilities without connections crossing? This is the classic "utilities problem" or the K_{3,3} bipartite graph problem. 2. **Exp

1. **Problem Statement:** Find the Cartesian product of the complete graph $K_3$ and the cycle graph $C_4$. 2. **Definitions:**

1. The problem is to understand the concept of extension edge Roman domination in graphs. 2. Roman domination in a graph is a function $f: V \to \{0,1,2\}$ assigning values to vert

1. The problem is to formulate a title that links the concept of edge Roman domination with the concept of neighborhood in graph theory. 2. Edge Roman domination is a variant of do

1. **Problem 1.1a:** Find all edges incident on vertex $v_1$. Edges incident on $v_1$ are those connected to $v_1$. Given:

1. The problem is to find the Minimum Spanning Tree (MST) of a connected, weighted graph using Prim's algorithm. 2. Prim's algorithm starts with a single vertex and grows the MST b

1. **Problem Statement:** Find the shortest path from vertex $a$ to vertex $z$ in the given graph using Dijkstra's algorithm. 2. **Graph Description:** Vertices: $a,b,c,d,e,f,g,z$.

1. **State the problem:** We need to show that the graphs in problems 31 and 32 do not have a Hamiltonian circuit. A Hamiltonian circuit is a cycle that visits every vertex exactly

1. **Problem Statement:** Determine if the given graphs have an Euler circuit based on their vertex degrees. 2. **Key Concept:** A graph has an Euler circuit if and only if it is c

1. **Problem Statement:** We have a graph with nodes $a$, $b$, and $c$ and edges:

1. **Nêu bài toán:** Cho đồ thị hướng biểu diễn các chuyến bay giữa các thành phố Boston, Newark, Miami, Detroit, Washington với số chuyến bay cụ thể.

1. Soal 1: Diberikan himpunan simpul $V = \{A, B, C, D\}$ dan himpunan sisi $E = \{\{A,B\}, \{A,C\}, \{B,D\}, \{C,D\}\}$. 2. Tentukan derajat setiap simpul. Derajat simpul adalah j

1. Masalah: Tentukan matriks bersisian (incidence matrix) untuk graf dengan 4 simpul dan 6 sisi. 2. Matriks bersisian adalah matriks yang menunjukkan hubungan antara simpul dan sis

1. The problem is to investigate Seymour's Second Neighborhood Conjecture, which states that in any finite directed graph, there exists a vertex whose second out-neighborhood is at