Algorithms Q&A - Recent questions in Graph Theory

Give an example of a graph that has 6 vertices, 9 edges, but does not have a clique on 3 vertices

Tue, 12 Dec 2023 03:28:42 +0000

Give an example of a graph that has 6 vertices, 9 edges, but does not have a clique on 3 vertices

Example of a graph that is connected, has articulation point, needs 4 colors, and does not have a 4-clique

Sat, 10 Dec 2022 21:36:51 +0000

Give an example of a graph that has all of the following properties. (Note that you need to give a single graph as the answer.)
(i) It is connected
(ii) It has one articulation point.
(iii) The graph needs at least 4 colors for a valid vertex coloring
(iv) The graph does not have a 4-clique (that is, a clique of 4 vertices) as a subgraph.

Give an example of a graph without an articulation point, a Hamiltonian cycle, or a 2-coloring

Mon, 09 Jul 2018 02:02:10 +0000

How to generate a random connected graph?

Sun, 15 Oct 2017 13:07:30 +0000

For running algorithms like Minimum Spanning Tree, we want to create a undirected weighted graph to test and the graph is very large like n vertices and m edges. How can we create a random graph with random number of vertices and edges, so that the graph is connected?

Graph with no 3-clique that needs at least 4 colors

Mon, 24 Jul 2017 12:47:46 +0000

Give an example of a graph that has the following properties. (Note that you need to give a single graph as the answer.)

The graph does not contain a triangle (that is, a clique of 3 vertices) as a subgraph.
Graph needs at least 4 colors for a proper vertex coloring

[If you think that such a graph is not possible, then prove that statement.]

Heap removing and adding

Fri, 07 Oct 2016 04:30:34 +0000

Now I have a heap or priority queue to maintain the max value of a series of numbers. Usually, if I want to pop the max value, fetching and removing the root number then putting the last number in the root and shifting down to balance; If I want to add a new number, adding the number at the end of the heap then shifting up to balance.

So my question: If I want to pop the max value and immediately add new value, could I just remove the root number and put the new one in the root then shifting down to balance? If it is wrong, why? If it is true, how to prove it?

Graph that has 10 vertices, is 3-colorable and number of edges is maximized

Tue, 23 Aug 2016 10:43:03 +0000

What is the maximum number of edges that can be present in a graph, that has 10 vertices, and has a valid vertex coloring with only 3 colors?

Graph Example n >= 6, delta >= 4, ki = 2

Sat, 20 Aug 2016 20:01:46 +0000

Give an example of a graph that contains at least 6 vertices, and each vertex has at least 4 neighbors, and the graph has a valid vertex coloring using only 2 colors.

Color this graph (n=10, m=15)

Tue, 16 Aug 2016 01:42:23 +0000

Can you color this graph, such that every vertex receives some color, no two adjacent vertices receive the same color, and number colors used is minimized?

symmetry + transitivity => reflexivity, or not?

Mon, 18 Jan 2016 18:30:12 +0000

Professor has posted a question that if one relation has the property of symmetry, such that if aRb, then bRa, and also transitivity, aRb, bRc so aRc, so can we say that because aRb and bRa, we can get aRa which is reflexivity?

Polynomial time solution for Constrained Version of Tetris Problem

Thu, 14 Jan 2016 07:41:04 +0000

Tetris is known to be NP-complete problem in general. However, in this constrained version, we only consider the straight tetrominoes, and further, we cannot rotate them.

We can make the general assumptions that the board is w x h (width x height). We can see the entire sequence of tetrominoes, which is n units long and only consists of straight tetrominoes (horizontal or vertical). We can place those tetrominoes wherever we like, but we cannot rotate them.

We receive k points for each row that becomes full. The full row is then cleared away.

We need to maximize the score before the height reaches h (not including the cleared rows).