#strongly connected graph and weakly connected graph

[codeshive]

CS 579 Homework 1 solved

1. Given the friendship graph below, calculate and plot the degree distribuFon of the graph. Be sure to label the plot axes. 2. Draw the graph specified in the adjacency matrix below. Is this graph connected? If yes, is it weakly connected or strongly connected? ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 7 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0⎦ ⎥ ⎥ ⎥ ⎥ ⎤ 3. Use Dijkstra’s or Prim’s Algorithm to…

[max1461]

Let G be a graph whose edges are colored either red or blue. A collection of nodes connected by blue edges is called an alliance. An alliance is weakly coherent if no two nodes are connected by a red edge. An alliance is strongly coherent if no two nodes are connected by a path consisting of an odd number of red edges. G is (weakly/strongly) coherent if all of its alliances are.

Proposition that might seem reasonable, but isn't true: all weakly coherent graphs are strongly coherent.

Counterexample: not the Syrian civil war, it's neither one.

I'm trying but I don't think I'm gonna understand what all the different factions in the syrian civil war are anytime soon

[theiteducation]

What is Connected & Strongly Connected Components in Graph

What is Connected & Strongly Connected Components in Graph

What is Connected & Strongly Connected Components in Graph. Let learn in Urdu & Hindi for the course CS502 and Cs702. A directed graph is called strongly connected if there is a path in each direction between each pair of vertices(u,v). It means if U vertex can reach to V vertex and V vertac can reach to Vertex U in the graph. That is, a path exists from the first vertex in the pair to the…

[mathematicianadda]

When does a graph have a circular orientation?

Call an oriented digraph $D=(V,A)$ circular when for all $\small x,y,z\in V$ if $(x,y)\in A$ and $(y,z)\in A$ then $(z,x)\in A$ or equivalently if $D$ is any oriented digraph whose arc set is a circular relation.

With that said, when does an undirected graph have a circular orientation?

I can prove these graphs are perfect i.e. graphs with circular orientations have identical clique and chromatic numbers for all their induced subgraphs, also I can show they are three-colorable and there exists a family of forbidden induced subgraphs which characterizes them. Its also easy to see these graphs look similar in their definition to comparability graphs i.e. graphs which have an orientation $D=(V,A)$ such that for all $x,y,z\in V$ if $(x,y)\in A$ and $(y,z)\in A$ then $(x,z)\in A$ (this last arc in the definition of circular orientations is flipped) also like graphs with circular orientations, graphs with transitive orientations are similarly perfect graphs. Now Gallai proved the following countable set $S$ of forbidden induced subgraph types characterized the comparability graphs:

$$\small S=\{(G_k)^{\complement}:1\leq k\leq 8\}\cup\{B_1^{\complement},B_2^{\complement}\}\cup\bigcup_{n=2}^{\infty}\{C_{2n+1},J_n,J'_{n+1},J''_n,(K_n)^{\complement},(C_{n+4})^{\complement},(L_{n-1})^{\complement},(L'_{n-1})^{\complement}\}$$

Where the indexed types $G,B,K,L,C,J,J',J''$ are each defined diagrammatically as follows:

Thus surely characterizing those graphs with circular orientations by a family of forbidden induced subgraphs can not be harder then this? So far ive reduced the problem to finding the graphs with a strongly-connected-circular-orientation. In fact one can prove that an oriented digraph $D$ is circular if and only if all of its weakly connected components are circular, moreover if any weakly connected oriented digraph $D$ is circular then one can prove either $D$ contains no paths of length two and thus underlies a connected bipartite graph or that $D$ is strongly connected. I can also prove if $D$ any strong circular oriented digraph then every dicycle in $D$ will have a length which is divisible by $3$ and every arc in $D$ will belong to a directed cycle of length $3$ in $D$. Thus the adajency matrix of $D$ will have an index of imprimitivity equal to $3$ which means if any graph $G$ has a circular orientation, then we must have as a corollary of this fact that:

$$\omega(G)=\chi(G)=\begin{cases}3\text{ if }G\text{ contains any triangle}\\2\text{ if }G\text{ contains no triangles}\end{cases}$$

Further it can be shown that the class of oriented circular digraphs is likewise closed under induced subdigraphs since if we let $\small F_1=(\{1,2,3\},\{(1,2),(2,3)\})$ and $\small F_2=(\{1,2,3\},\{(1,2),(2,3),(1,3)\})$ then any oriented digraph $D$ is circular iff no induced subdigraph of $D$ is isomorphic to $F_1$ or $F_2$. Also if $D$ is any strong circular oriented digraph then for (not necessarily distinct) $\small u,v\in V(D)$ if we define $d(u,v)$ to be equal to the length of any shortest walk from $u$ to $v$ in the digraph $D$ then we can write:

$$d(u,v)=\begin{cases}3\text{ if }(v,u)\not\in E(D)\text{ and }(u,v)\not\in E(D)\\2\text{ if }(v,u)\in E(D)\\1\text{ if }(u,v)\in E(D)\end{cases}$$

As there is a partition $\{A,B,C\}$ of $V(D)$ such that $\small E(D)\subseteq (A\times B)\cup (B\times C)\cup (C\times A)$. Now I suspect I have gone overboard with this and that someone familiar in this area could tell me how to characterize them, would someone please help me? I'd really appreciate it. I have been unable to find any in depth study of circular relations apart from people commonly noting that equivalence relations can be characterised as reflexive circular relations, as well as a few articles on computer science, and this paper: https://www.jstor.org/stable/24339780?seq=1 which keeps showing up in most of my searches. This leads me to think these are either not very important or they go by a different name or perhaps even characterizing them is so trivial no one has cared to write about it in a paper. Can someone clarify for me?

from Hot Weekly Questions - Mathematics Stack Exchange from Blogger https://ift.tt/2QYwmvr

[kaliforniaco]

Check if a graph is Strongly, Unilaterally or Weakly connected https://t.co/f1bweYZyGB

Check if a graph is Strongly, Unilaterally or Weakly connected https://t.co/f1bweYZyGB

— Dave Epps (@dave_epps) July 5, 2020

from Twitter https://twitter.com/dave_epps

[craigbrownphd]

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[fullgrade-blog]

C++ code: Different Graph connected

Using VS and C++. I will provide a Source file which is a Graph class. The requirement is to add 4 methods:

· bool isStronglyConnected() – returns true if the graph instance is strongly connected.

· bool isWeaklyConnected() – returns true if the graph instance is weakly connected.

· int largestWeaklyConnectedSubGraph()– returns the number of vertices in the weakly connected sub-graph with the…