GATE Overflow - Recent questions tagged graph-connectivity
http://gateoverflow.in/tag/graph-connectivity
Powered by Question2Answer2 - connected graph
http://gateoverflow.in/130141/2-connected-graph
<p>For a <strong>regular graph</strong> how much large the value of degree (for each vertices) should be such that the graph is $2$ - connected. (vertex wise).</p>
<p>I did in this way :</p>
<p>$\begin{align*} &\quad \kappa(G) \leq \frac{2\cdot e}{n} \qquad \text{ where } \kappa(G) = \text{ vertex connectivity } \\ &\Rightarrow 2 \leq \frac{2\cdot e}{n} \\ &\Rightarrow n \leq e \\ &\Rightarrow n \leq \frac{\sum \left ( d_i \right )}{2} \\ &\Rightarrow n \leq \frac{n \cdot d}{2} \\ &\Rightarrow d \geq 2 \\ \end{align*}$</p>
<p>The above case can be realized by thinking of a <strong>cycle graph</strong> of $n$ vertices.</p>
<p>But in the following case :</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=15360590681287688332"></p>
<p>This graph is 3 regular and not 2- connected although $d \geq 2$ is satisfied.</p>
<p>Why this $d \geq 2$ is trivial and not working in some cases ?</p>Graph Theoryhttp://gateoverflow.in/130141/2-connected-graphFri, 19 May 2017 07:12:05 +0000GATE Graph Theory
http://gateoverflow.in/129447/gate-graph-theory
Let G = (V, E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G?<br />
<br />
( A ) G1 = (V, E1) where E1 = {(u, v) | (u, v) ∉ E}<br />
( B ) G2 = (V, E2) where E2 = {(u, v) | (v, u) ∉ E}<br />
( C ) G3 = (V, E3) where E3 = {(u, v) | there ish a path of length ≤ 2 from u to v in E}<br />
( D ) G4 = (V4, E) where V4 is the set of vertices in G which are not isolated<br />
<br />
Can anyone give a detailed answer to this question, please? :)Graph Theoryhttp://gateoverflow.in/129447/gate-graph-theoryFri, 12 May 2017 19:24:26 +0000Graph Theory
http://gateoverflow.in/129355/graph-theory
algorithm to find more than one path between any two vertices of a graph G=(V,E) , with a complexity of O(VE) ?Graph Theoryhttp://gateoverflow.in/129355/graph-theoryFri, 12 May 2017 08:04:59 +0000graph theory
http://gateoverflow.in/124669/graph-theory
A graph consists of only one vertex,which is isolated ..Is that graph<br />
<br />
A) a complete graph ???<br />
<br />
B) a clique???<br />
<br />
C) connected graph ???<br />
<br />
Please explain your answer ...Graph Theoryhttp://gateoverflow.in/124669/graph-theoryFri, 07 Apr 2017 17:38:26 +0000graph theory
http://gateoverflow.in/121379/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14124213158983667542"></p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=5624474628795835148"></p>Graph Theoryhttp://gateoverflow.in/121379/graph-theorySun, 12 Mar 2017 17:52:08 +0000graph theory
http://gateoverflow.in/121304/graph-theory
<p><strong>chromatic number of a graph <= ( maxdegree of the graph ) + 1 </strong></p>
<p>can somebody explain how ?</p>Graph Theoryhttp://gateoverflow.in/121304/graph-theorySat, 11 Mar 2017 16:53:30 +0000graph theory
http://gateoverflow.in/121303/graph-theory
A graph with n vertices and 0 edges.can this graph be called as Bipartite ? i mean can we simply partition the n vertices into two sets of vertices such that there is no edge within the set as well there is no edge between the two sets and say it as a Bipartite graph ?Graph Theoryhttp://gateoverflow.in/121303/graph-theorySat, 11 Mar 2017 16:50:02 +0000graph theory
http://gateoverflow.in/121282/graph-theory
State TRUE or FALSE.<br />
<br />
The chromatic number of a Bi-partite graph is ALWAYS 2.Graph Theoryhttp://gateoverflow.in/121282/graph-theorySat, 11 Mar 2017 10:57:10 +0000graph theory
http://gateoverflow.in/121278/graph-theory
The cardinality of the vertex-cut ( seperating set ) of a complete graph with n vertices is ___Graph Theoryhttp://gateoverflow.in/121278/graph-theorySat, 11 Mar 2017 10:16:37 +0000graph theory
http://gateoverflow.in/121240/graph-theory
In a Bipartite graph,the size of the maximum matching is equal to the size of the minimum vertex cover ...can somebody prove this logically ?Graph Theoryhttp://gateoverflow.in/121240/graph-theoryFri, 10 Mar 2017 19:06:22 +0000graph theory
http://gateoverflow.in/121147/graph-theory
<p>The number of <strong>independent sets</strong> in a complete graph with n vertices is ____</p>Graph Theoryhttp://gateoverflow.in/121147/graph-theoryFri, 10 Mar 2017 10:03:49 +0000graph theory
http://gateoverflow.in/121111/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=16494923851889305305"></p>
<p>can somebody explain the logic behind this theorem ?</p>Graph Theoryhttp://gateoverflow.in/121111/graph-theoryThu, 09 Mar 2017 21:25:28 +0000graph theory
http://gateoverflow.in/121050/graph-theory
<p> </p>
<p>Find </p>
<p>1) Vertex connectivity </p>
<p>2) Edge connectivity </p>
<p>3) Is it a seperable graph ? If so then find the cut-vertex </p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11584695799903342456"></p>Graph Theoryhttp://gateoverflow.in/121050/graph-theoryThu, 09 Mar 2017 10:25:56 +0000graph theory
http://gateoverflow.in/121042/graph-theory
<p> Find </p>
<p>1) Vertex connectivity </p>
<p>2) Edge connectivity </p>
<p>3) Is it a seperable graph ? If so then find the cut-vertex </p>
<p>4) Is {v<sub>1</sub>,v<sub>2</sub>,v<sub>5</sub>} a cut-set ?</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=38764623861261551"></p>Graph Theoryhttp://gateoverflow.in/121042/graph-theoryThu, 09 Mar 2017 10:10:12 +0000graph theory
http://gateoverflow.in/120989/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=13344674821615170849"></p>Graph Theoryhttp://gateoverflow.in/120989/graph-theoryWed, 08 Mar 2017 20:35:19 +0000graph theory
http://gateoverflow.in/120988/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=16810418372386685647"></p>Graph Theoryhttp://gateoverflow.in/120988/graph-theoryWed, 08 Mar 2017 20:31:38 +0000ISI 2015 PCB C3
http://gateoverflow.in/120885/isi-2015-pcb-c3
For a positive integer n, let G = (V, E) be a graph, where V = {0,1}^n, i.e., V is the set of vertices has one to one correspondence with the set of all n-bit binary strings and E = {(u,v) | u, v belongs to V, u and v differ in exactly one bit position}.<br />
<br />
i) Determine size of E<br />
<br />
ii) Show that G is connectedGraph Theoryhttp://gateoverflow.in/120885/isi-2015-pcb-c3Wed, 08 Mar 2017 07:22:42 +0000Graph Theory Me workbook
http://gateoverflow.in/115573/graph-theory-me-workbook
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=223024569878688878"></p>
<p>How S2 is correct ,I can have more than n-k edges like if n=7 and k=3 ,then K1(a-b-c-d-e) k2(f() k2(g).K1,k2,k3 are different compoinents i assumes,Now in K1 i can add one more edge between a to c or a to d and still it will be simple graph and it will have 3 components?Please help</p>Set Theory & Algebrahttp://gateoverflow.in/115573/graph-theory-me-workbookSat, 04 Feb 2017 10:58:17 +0000graph theory
http://gateoverflow.in/100206/graph-theory
Assumed undirected graph G is connected. G has 6vertices and 10 edges. Find<br />
the minimum number of edges whose deletion from graph G is always guarantee<br />
that it will become disconnected.Graph Theoryhttp://gateoverflow.in/100206/graph-theoryWed, 04 Jan 2017 02:27:09 +0000graph theory
http://gateoverflow.in/97768/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=11696266218365235197"></p>Graph Theoryhttp://gateoverflow.in/97768/graph-theoryWed, 28 Dec 2016 19:18:36 +0000graph
http://gateoverflow.in/92032/graph
Which of the following statements is/are TRUE?<br />
[P] Every disconnected graph has an isolated vertex<br />
[Q] A graph is connected if and only if some vertex is connected to all other vertices<br />
[R] The edge set of every closed trail can be partitioned into edge sets of cycles<br />
[S] If a maximal trail in a graph is not closed, then its endpoints have odd degreeGraph Theoryhttp://gateoverflow.in/92032/graphMon, 12 Dec 2016 16:20:24 +0000Graph theory
http://gateoverflow.in/92030/graph-theory
proof :- A connected graph any two paths of maximum length share at least one vertexGraph Theoryhttp://gateoverflow.in/92030/graph-theoryMon, 12 Dec 2016 16:09:47 +0000graph theory
http://gateoverflow.in/84795/graph-theory
Which of the following statements is/are TRUE?<br />
[P] Every disconnected graph has an isolated vertex<br />
[Q] A graph is connected if and only if some vertex is connected to all other vertices<br />
[R] The edge set of every closed trail can be partitioned into edge sets of cycles<br />
[S] If a maximal trail in a graph is not closed, then its endpoints have odd degreeGraph Theoryhttp://gateoverflow.in/84795/graph-theoryTue, 22 Nov 2016 12:55:21 +0000Doubt: Graph Theory
http://gateoverflow.in/80069/doubt-graph-theory
When say that with n vertices there are total 2^(n(n-1)/2) connected/disconnected graph possible, in this case we are assuming that vertices are labelled, right??<br />
<br />
<br />
<br />
Is there any formula to count number of connected/disconnected graphs possible with n unlabeled vertices?Graph Theoryhttp://gateoverflow.in/80069/doubt-graph-theoryTue, 08 Nov 2016 08:41:14 +0000Kerala PSC AP Exam
http://gateoverflow.in/77267/kerala-psc-ap-exam
The maximum number of edges in an acyclic undirected graph with n vertices<br />
<br />
A) n - 1<br />
<br />
B) n<br />
<br />
C) n +1<br />
<br />
D) 2n -1DShttp://gateoverflow.in/77267/kerala-psc-ap-examThu, 27 Oct 2016 22:23:07 +0000Graph theory
http://gateoverflow.in/76955/graph-theory
Decomposition of complete graph into cycles through all vertices.<br />
Continuing explanation [here][1],<br />
Next explanation is given as<br />
<br />
for $n=5$ , $n=7$, it suffices to use cycles formed by traversing the<br />
vertices with constant difference:$\left(0,1,2,3,4\right)$,$\left(0,2,4,1,3\right)$ for $n=5 $<br />
<br />
and<br />
<br />
$\left(0,1,2,3,4,5,6\right)$,$\left(0,3,6,2,5,1,4\right)$ for $n=7 $<br />
<br />
Not getting how <br />
$\left(0,1,2,3,4\right)$,$\left(0,2,4,1,3\right)$ and $\left(0,1,2,3,4,5,6\right)$,$\left(0,3,6,2,5,1,4\right)$ is coming from !!!!!<br />
<br />
Please help me out!!<br />
<br />
[1]: <a href="http://math.stackexchange.com/questions/1985647/decomposition-of-complete-graph-into-cycles-through-all-vertices/1985652#1985652" rel="nofollow" target="_blank">http://math.stackexchange.com/questions/1985647/decomposition-of-complete-graph-into-cycles-through-all-vertices/1985652#1985652</a>Graph Theoryhttp://gateoverflow.in/76955/graph-theoryWed, 26 Oct 2016 17:59:26 +0000Graph connectivity
http://gateoverflow.in/73128/graph-connectivity
<p><img alt="Loading Question" src="https://d2190hpfa85jkd.cloudfront.net/q/aa27563a2b91fc7079e234e2873059f7.jpg"></p>Graph Theoryhttp://gateoverflow.in/73128/graph-connectivityMon, 10 Oct 2016 18:23:34 +0000Gatebook
http://gateoverflow.in/63846/gatebook
<h2>Let A has n vertices. If Ā is connected graph then the maximum number of edges that A can have is</h2>
<h3>a) (n-1)(n-2)/2
<br>
b) n(n-1)/2
<br>
c) n-1
<br>
d) n</h3>Mathematical Logichttp://gateoverflow.in/63846/gatebookFri, 19 Aug 2016 17:03:49 +0000Graph Theory
http://gateoverflow.in/62888/graph-theory
consider the following statement-:<br />
<br />
1.If a graph has Euler circuit then it is Strongly Connected graph.<br />
<br />
2.If a graph has Euler path(but not Euler circuit) then it is Strongly Connected graph.<br />
<br />
3.If a graph has Euler circuit then it is Weakly Connected graph.<br />
<br />
4.If a graph has Euler path(but not euler circuit) then it is Weakly Connected graph.<br />
<br />
Which statement is true with proper explanation.Graph Theoryhttp://gateoverflow.in/62888/graph-theoryFri, 12 Aug 2016 10:09:13 +0000Graph theory
http://gateoverflow.in/62654/graph-theory
complement of a complete bipartite graph Km,n .please provide a figure for explanation.Graph Theoryhttp://gateoverflow.in/62654/graph-theoryWed, 10 Aug 2016 17:36:35 +0000UGCNET-Sep2013-II-20
http://gateoverflow.in/49362/ugcnet-sep2013-ii-20
<p>Consider the following statements:</p>
<ol style="list-style-type: upper-roman;">
<li>A graph in which there is a unique path between every pair of vertices is a tree.</li>
<li>A connected graph with e=v-1 is a tree</li>
<li>A connected graph with e=v-1 that has no circuit is a tree</li>
</ol>
<p>Which one of the above statements is/are true?</p>
<ol style="list-style-type: upper-alpha;">
<li>I and III</li>
<li>II and III</li>
<li>I and II</li>
<li>All of the above</li>
</ol>
DShttp://gateoverflow.in/49362/ugcnet-sep2013-ii-20Thu, 09 Jun 2016 13:14:47 +0000CMI2011-B-02b
http://gateoverflow.in/47081/cmi2011-b-02b
<p style="line-height: 20.8px;">Let $G$ be a connected graph. For a vertex $x$ of $G$ we denote by $G−x$ the graph formed by removing $x$ and all edges incident on $x$ from $G$. $G$ is said to be good if there are at least two distinct vertices $x, y$ in $G$ such that both $G − x$ and $G − y$ are connected.</p>
<ol style="line-height: 20.8px; list-style-type: lower-roman;">
<li>Given a good graph, devise a linear time algorithm to find two such vertices.</li>
</ol>
<p style="line-height: 20.8px;"> </p>
Graph Theoryhttp://gateoverflow.in/47081/cmi2011-b-02bFri, 27 May 2016 13:37:07 +0000CMI2011-B-02a
http://gateoverflow.in/46203/cmi2011-b-02a
<p>Let $G$ be a connected graph. For a vertex $x$ of $G$ we denote by $G−x$ the graph formed by removing $x$ and all edges incident on $x$ from $G$. $G$ is said to be good if there are at least two distinct vertices $x, y$ in $G$ such that both $G − x$ and $G − y$ are connected.</p>
<ol style="list-style-type: lower-roman;">
<li>Show that for any subgraph $H$ of $G$, $H$ is good if and only if $G$ is good.</li>
</ol>
Set Theory & Algebrahttp://gateoverflow.in/46203/cmi2011-b-02aThu, 19 May 2016 11:59:10 +0000CMI2011-A-07
http://gateoverflow.in/46194/cmi2011-a-07
<p>Let $G=(V, E)$ be a graph. Define $\bar{G}$ to be $(V, \bar{E})$, where for all $u, \: v \: \in V \: , (u, v) \in \bar{E}$ if and only if $(u, v) \notin E$. Then which of the following is true?</p>
<ol style="list-style-type: upper-alpha;">
<li>$\bar{G}$ is always connected.</li>
<li>$\bar{G}$ is connected if $G$ is not connected.</li>
<li>At least one of $G$ and $\bar{G}$ connected.</li>
<li>$G$ is not connected or $\bar{G}$ is not connected</li>
</ol>
Graph Theoryhttp://gateoverflow.in/46194/cmi2011-a-07Thu, 19 May 2016 11:40:12 +0000CMI2010-B-02
http://gateoverflow.in/46129/cmi2010-b-02
<p><span style="line-height: 20.8px;">Let $G$ be a graph in which each vertex has degree at least $k$. Show that there is a path of length $k$ in $G$—that is, a sequence of $k+1$ distinct vertices $v_0, v_1, \dots v_k$ such that for $0 \leq i < k,$ $v_i$ is connected to $v_{i+1}$ in $G$.</span></p>
Graph Theoryhttp://gateoverflow.in/46129/cmi2010-b-02Thu, 19 May 2016 06:18:54 +0000Connectivity Graph Theory
http://gateoverflow.in/45037/connectivity-graph-theory
Which of the following statements are true . Please explain why each statement is true/false..<br />
<br />
S1 : If a simple graph G is not connected then it's complement G is not connected<br />
<br />
S2 : If a simple graph G is connected them it's complement G is not connected<br />
<br />
S3 : A simple graph with n vertices is necessarily connected if min degree of a vertex = (n-1)/2<br />
<br />
S4 : If a simple graph has exactly two vertices of odd degree then there exists a path between two vertices of odd degreeGraph Theoryhttp://gateoverflow.in/45037/connectivity-graph-theoryFri, 06 May 2016 18:07:19 +0000GATE2015-1_54
http://gateoverflow.in/8364/gate2015-1_54
Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is_______________.Graph Theoryhttp://gateoverflow.in/8364/gate2015-1_54Sat, 14 Feb 2015 04:55:13 +0000GATE2015-2_50
http://gateoverflow.in/8252/gate2015-2_50
<p>In a connected graph, a bridge is an edge whose removal disconnects the graph. Which one of the following statements is true?</p>
<p> </p>
<ol style="list-style-type:upper-alpha">
<li>A tree has no bridges</li>
<li>A bridge cannot be part of a simple cycle</li>
<li>Every edge of a clique with size ≥ 3 is a bridge (A clique is any complete subgraph of a graph)</li>
<li>A graph with bridges cannot have cycle</li>
</ol>
Graph Theoryhttp://gateoverflow.in/8252/gate2015-2_50Fri, 13 Feb 2015 09:42:54 +0000GATE2005-IT-56
http://gateoverflow.in/3817/gate2005-it-56
<p>Let G be a directed graph whose vertex set is the set of numbers from 1 to 100. There is an edge from a vertex i to a vertex j iff either j = i + 1 or j = 3i. The minimum number of edges in a path in G from vertex 1 to vertex 100 is</p>
<ol style="list-style-type: upper-alpha;">
<li>4</li>
<li>7</li>
<li>23</li>
<li>99</li>
</ol>Graph Theoryhttp://gateoverflow.in/3817/gate2005-it-56Tue, 04 Nov 2014 03:06:29 +0000GATE2004-IT-37
http://gateoverflow.in/3680/gate2004-it-37
<p>What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?</p>
<ol style="list-style-type: upper-alpha;">
<li>10</li>
<li>11</li>
<li>18</li>
<li>19</li>
</ol>Graph Theoryhttp://gateoverflow.in/3680/gate2004-it-37Sun, 02 Nov 2014 09:32:47 +0000GATE2004-IT-5
http://gateoverflow.in/3646/gate2004-it-5
<p>What is the maximum number of edges in an acyclic undirected graph with n vertices?</p>
<ol style="list-style-type: upper-alpha;">
<li>n-1</li>
<li>n</li>
<li>n+1</li>
<li>2n-1</li>
</ol>Graph Theoryhttp://gateoverflow.in/3646/gate2004-it-5Sun, 02 Nov 2014 03:17:00 +0000GATE2006-IT-11
http://gateoverflow.in/3550/gate2006-it-11
<p>If all the edge weights of an undirected graph are positive, then any subset of edges that connects all the vertices and has minimum total weight is a</p>
<ol style="list-style-type: upper-alpha;">
<li>Hamiltonian cycle</li>
<li>grid</li>
<li>hypercube</li>
<li>tree</li>
</ol>Graph Theoryhttp://gateoverflow.in/3550/gate2006-it-11Fri, 31 Oct 2014 06:41:28 +0000GATE2008-IT-27
http://gateoverflow.in/3317/gate2008-it-27
<p>G is a simple undirected graph. Some vertices of G are of odd degree. Add a node v to G and make it adjacent to each odd degree vertex of G. The resultant graph is sure to be</p>
<ol style="list-style-type: upper-alpha;">
<li>regular</li>
<li>complete</li>
<li>Hamiltonian</li>
<li>Euler</li>
</ol>Graph Theoryhttp://gateoverflow.in/3317/gate2008-it-27Tue, 28 Oct 2014 13:06:37 +0000GATE1995_1.25
http://gateoverflow.in/2612/gate1995_1-25
The minimum number of edges in a connected cyclic graph on $n$ vertices is:<br />
<br />
(a) $n-1$<br />
(b) $n$<br />
(c) $n+1$<br />
(d) None of the aboveGraph Theoryhttp://gateoverflow.in/2612/gate1995_1-25Wed, 08 Oct 2014 23:23:31 +0000GATE1993_8.1
http://gateoverflow.in/2299/gate1993_8-1
<p>Consider a simple connected graph $G$ with $n$ vertices and $n$ edges $(n > 2)$. Then, which of the following statements are true?</p>
<ol style="list-style-type: upper-alpha;">
<li>$G$ has no cycles</li>
<li>The graph obtained by removing any edge from $G$ is not connected</li>
<li>$G$ has at least one cycle</li>
<li>The graph obtained by removing any two edges from $G$ is not connected</li>
<li>None of the above </li>
</ol>
Graph Theoryhttp://gateoverflow.in/2299/gate1993_8-1Tue, 30 Sep 2014 00:15:21 +0000GATE2014-2-3
http://gateoverflow.in/1955/gate2014-2-3
The maximum number of edges in a bipartite graph on 12 vertices is____Graph Theoryhttp://gateoverflow.in/1955/gate2014-2-3Sun, 28 Sep 2014 10:19:01 +0000GATE1999_1.15
http://gateoverflow.in/1468/gate1999_1-15
<p>The number of articulation points of the following graph is</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=6004385987067299973" style="height:156px; width:336px"></p>
<ol style="list-style-type:upper-alpha">
<li>0</li>
<li>1</li>
<li>2</li>
<li>3</li>
</ol>
<p> </p>
Graph Theoryhttp://gateoverflow.in/1468/gate1999_1-15Tue, 23 Sep 2014 23:29:39 +0000