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# All Questions

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### How to correctly describe this action, deleting an edge that “shortcut” some vertices

Haven't written a proof in years, not sure how to describe an algorithm like this ? Let us what we have a graph. like this below: 1). How to describe edge removal of{ (0, 1),(3,4), (1,2) }done in ...
214 views

### Proof that G is a Tree After DFS and BFS form the same tree T [closed]

Let G be a connected, undirected graph containing some vertex s. let's say that BFS and DFS are both run on G starting at s and that the breadth first search and depth first search ...
67 views

### Intuitive proof for a tree with n nodes, has n-1 edges

I am interested in an intuitive proof for "any binary tree with $n$ nodes has $n-1$ edges", that goes beyond proof by strong induction.
84 views

### Alternate proof of the Caro-Wei theorem for lower bounding the independence number

Let $G$ be a graph on $n$ vertices whose degree sequence is $d_1,d_2,...,d_n$. Let $\alpha(G)$ denote the size of maximum independent set of $G$, i.e., the size of a maximum subset of vertices of $G$ ...
69 views

### Proof By Contradiction - Hamiltonian Paths and Cycles

Was hoping if anyone had any way to prove the following claim using proof by contradiction Let $G = (V, E)$ be a simple graph with at least one vertex, and let $G'$ be the graph formed by adding a ...
23 views

### There could not be an edge from u to v in a DAG, if w is before v in a topological order

I am trying to prove that given a DAG. There exists a valid topological ordering that has v in front of u iff there is no path from u to v. The proof is related to the fact that reverse DFS post ...
185 views

### lower bound proof with adversary argument

We have to run a song on a Walkman, for that we need 2 full batteries. Let's say we have a mixed set of 30 batteries (15 are empty and and 15 are full) and then only way to test if the battery is full ...
75 views

### If a Triple Graph Grammar rule counts as a Mathematical Proof

I am intrigued by Triple Graph Grammars (TGG) as a potential for formal mathematical proof. Triple Graph Grammars (TGGs) are a technique for defining the correspondence between two different ...
79 views

### how to prove original intervals and canonical form of intervals have the same interval graph

According to this paper,a family of intervals is said to be canonical if the coordinates of the endpoints of the intervals are distinct integers between 1 and 2n where n is the number of intervals. ...
135 views

### Find, with proof, the number of distinct graphs with the vertex set $V = \{1, 2, \ldots,n\}$ [closed]

Let $n \in \mathbb{Z^+}$. Find, with proof, the number of distinct graphs with the vertex set $V = \{1, 2, \ldots,n\}$. We say two such graphs are distinct if one of them has an edge $(u, v)$ and the ...
94 views

### Prove that at least as many edges as vertices implies a cycle

I am EXTREMELY confused on where to start with this problem. We recently just started learning about graph theory and I don't know where to begin. Prove that in a connected graph G with $p$ ...
412 views

### How to show that two vertices in a connected component are in the same set? (bi conditional)

Show that after all edges are processed by CONNECTED-COMPONENTS, two vertices are in the same connected component if and only if they are in the same set. The CONNECTED-COMPONENTS algorithm is the ...
134 views

### What do we mean when we say an edge (u,v) connects some component to other component in forest G = (V,A)

Let H = (V,E) be a connected, undirected graph. Let A be a subset of E. Let C = (W , F) be a connected component (tree) in the forest G = (V,A). Let (u,v) be an edge connecting C to some other ...
2k views

### Why is T not a minimum spanning tree of G?

The Problem: Let T be a tree constructed by Dijkstra's algorithm in the process of solving the single source shortest-paths problem for a weighted connected graph G.    a. True of false:...
1k views

### How to draw a graph to disprove this statement?

The Problem: Indicate whether the following statements are true or false: a. If e is a minimum-weight edge in a connected weighted graph, it must be among edges of at least one minimum ...
153 views

### Birkhoff-von Neumann theorem for bistochastic digraphs

A weighted digraph (with loops) is bistochastic, iff the weights are non-negative, for all non-sink nodes, the sum of the edge weights of the out-edges is $1$, and for all non-source nodes, the sum ...
179 views

### Maximum flow problem with non-zero lower bound

Given $G = (V,E )$ a directed graph, if $X \subseteq V$ we write \begin{align*} \delta ^{+}(X) &= \{ xy\in E \mid x \in X, y\in V - X \} \\ \delta ^{-}(X) &= \delta ^{+}(V -...
222 views

### Unclear about proof for unique MST given graph G with distinct weights

http://homepages.math.uic.edu/~leon/cs-mcs401-s08/handouts/mst.pdf I have some trouble understanding the proof above. I understand that we assuming two MSTs, T and T', and an edge e that is the ...
86 views

### if (dis)proving a conjecture on graph theory can be done just by a counter example then can every (dis)proof be mapped actually to a counter-example?

Suppose we have a conjecture on graph theory that can be (dis)proved by means of a counter example, then, is it true that every alternative (dis)proof of the conjecture can be mapped to a counter ...
136 views

### Choose $n/2$ vertices and guarantee $3/4$ of edges are accounted for proof

Give a polynomial-time algorithm that finds ceil(V/2) vertices that collectively account for at least three-fourths (3/4) of the edges in an arbitrary undirected graph. The algorithm I have come up ...
1k views

### Proof of Ramsey's theorem: the number of cliques or anti cliques in a graph

Ramsey's theorem states that every graph with $n$ nodes contains either a clique or an independent set with at least $\frac{1}{2}\log_2 n$ nodes. I tried to look it up at a few places (including ...
1k views

### r-regular graph and hamiltonian path

I am having some issues proving a problem I am working on. I have been sketching out examples but the proof is not jumping out at me. Question: Let $G = (V,E)$ be an undirected $r$-regular graph (...
2k views

### Acyclic Tournament Digraphs and Hamiltonian Paths

I am studying MIT OCW lecture notes but they do not have solutions for the following problem. Directed Acyclic Tournaments In a round-robin tournament, every two distinct players play against each ...
2k views

I have this simple 'assignment' problem: We have a set of agents $A = \{a_1, a_2, \dotso, a_n\}$ and set of tasks $T= \{t_1, t_2, \dotso, t_m\}$. Note that $m$ is not necessarily equal to $n$. Unlike ...
2k views

### How can I prove that a complete binary tree has $\lceil n/2 \rceil$ leaves?

Given a complete binary tree with $n$ nodes. I'm trying to prove that a complete binary tree has exactly $\lceil n/2 \rceil$ leaves. I think I can do this by induction. For $h(t)=0$, the tree is ...
Let $G = (V,E)$ be a graph having $n$ vertices, none of which are isolated, and $n−1$ edges, where $n \geq 2$. Show that $G$ contains at least two vertices of degree one. I have tried to solve this ...
### Proving a binary tree has at most $\lceil n/2 \rceil$ leaves
I'm trying to prove that a binary tree with $n$ nodes has at most $\left\lceil \frac{n}{2} \right\rceil$ leaves. How would I go about doing this with induction? For people who were following in the ...