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Questions tagged [connected]

27 questions
8
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1 answer

Tarjan's SCC : example showing necessity of lowlink definition and calculation rule?

Several questions (1, 2) have been asked about this topic already but I am trying to be more specific. In Tarjan's SCC algorithm, the calculation of lowlink when encountering a vertex which is already on the stack is // It says w.index not…
ysalmon
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5
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1 answer

Algorithm to determine which vertices/edges would disconnect undirected graph if removed

Is anyone aware of an algorithm to determine which vertices/edges would disconnect an undirected graph if removed? For all vertices/edges. Of course I could run a BFS for each vertex and for each edge to test if the remaining graph would be…
xblax
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4
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1 answer

Maximum set of equalities, subject to some inequalities

I have $n$ variables $x_1,\dots,x_n$. I'm given a set $E$ of equalities (each of the form $x_i=x_j$ for some $i,j$) and a set $I$ of inequalities (each of the form $x_i \ne x_j$ for some $i,j$). I want to find a maximum-size subset $E' \subseteq…
D.W.
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4
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3 answers

How to prove that the dual of the dual of a connected planar graph $G$ is isomorphic to $G$?

I find in many references the fact that if $G$ is a connected planar graph, then for any embedding, $G^{**} \cong G$. However, all those references either say that this fact is trivial, or give the proof as an exercice. I am ok that this is clear on…
Nathaniel
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3
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1 answer

Determine whether removing a node on a graph will disconnect the graph

I have an undirected unweighted graph with a single component. I want to know if removing a given node will disconnect the graph. I know that I can remove the edges to the node and then use DFS to get the number of components, however this seems…
k-a-v
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2
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1 answer

number of connected subgraphs of $G$ with at most $ k>0$ vertices

Suppose we have an undirected graph $G=(V,E)$ with $n$ vertices and with max degree of $d>0$. I need to prove that number of connected subgraphs of $G$ with at most $n\geq k>0$ vertices is $M\leq n\cdot d^{2k}$ My approach was by induction. For…
ms_stud
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2
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3 answers

Connected components of the graph on $[n]$ in which $i,j$ are connected if $\mathrm{gcd}(i,j) > g$

I recently got asked the following question: A set of $n$ cities are numbered from 1 to $n$. Given a positive integer $g$, two cities are connected if their greatest common divisor is greater than $g$. The number $n$ may be as large as…
LiavK
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2
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Counting the number of connected components in a dynamic plane graph

I'm working on the following problem: let $G = (V, E)$ be a connected, planar graph. Our goal is to find a $d$-partition of $G$, $P = \{V_1, \ldots, V_d\}$, such that $G[V_i]$ is connected, and $\min_{V_i \in P} |V_i|$ is maximized. (That is, the…
rohan
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2
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1 answer

Strongly connected and completely specified Moore equivalent of a Mealy Machine

Problem: Prove that if a Mealy machine is strongly connected and completely specified, the corresponding Moore machine will also be strongly connected and completely specified. My approach so far: As the Mealy Machine is completely specified, we…
someone1
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2
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1 answer

Algorithm for finding connected components checking as few edges as possible

Is there a good algorithm to find connected components in undirected graphs with at the lowest possible costs given as the total weight of the edges being checked?
sortega
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2
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0 answers

Testing hidden connectivity of vertex pairs in graph

I have a problem from graph theory for which I could need some theoretical background and, if possible, ideas for an algorithm. Suppose we have a "usual" graph $G =(V,E)$ with finitely many vertices $V$ and edges $E$, at most one edge between two…
Jürgen Böhm
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2
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1 answer

Boolean constraints for a connected component of a graph

Suppose I have an undirected graph $G=(V,E)$, and boolean variables $x_v$ (one for each vertex $v \in V$). These variables select a subset $S \subseteq V$ of vertices, namely the vertices $S=\{v \mid x_v\}$ whose corresponding boolean variable is…
D.W.
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2
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1 answer

Select fixed-size connected induced subgraphs in a graph

I have a connected graph with $nk$ vertices, and would like to select $n$ disjoint induced subgraphs with $k$ vertices such that each subgraph is connected, selecting one out of all possible solutions at random. It doesn't necessarily need to be…
pommicket
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2
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1 answer

Stretegy to find the min expected cost on a series graph with edge probability pi and search cost ci

In a series graph, each edge $e_i$ exists with probability $p_i$. And if you want to examine the existence of edge $e_i$, it will cost you $c_i$. I want to test the connectivity between source $s$ and destination $d$ with the minimum expexted…
aweftr
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2
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1 answer

if a graph G(V,E) is connected $|E|\geq|V|-1$

If a graph G(V,E) is connected the number of edges is at least the number of Vertices-1. It is pretty evident if you think about it but how do i prove it formally?
toploz
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