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Questions tagged [graph-connectivity]

For questions related to the vertex-connectivity or edge-connectivity of graphs or networks: the minimum number of vertices (respectively edges) that need to be deleted to disconnect the graph.

The connectivity of a graph is a measure of the graph's resilience as a network. The two most important measures of connectivity are:

  • The vertex connectivity $\kappa(G)$ of a graph $G$: the minimum number of vertices that need to be deleted in order to disconnect the graph.
  • The edge connectivity $\kappa'(G)$ of a graph $G$: the minimum number of edges that need to be deleted in order to disconnect the graph.

(As an exception to the usual definition, we let $\kappa(K_n) = n-1$ and $\kappa'(K_1) = 0$, since there is not such a minimum number for these graphs.)

There are also variants of these definitions for directed graphs, for weighted graphs, and for connectivity between two specific vertices.

A key result related to the connectivity of a graph is Menger's theorem.

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Euler's formula doesn't work for null graph?

Given the null graph with no edges or vertices, we have a connected planar graph as no edges cross when this graph is drawn in the plane, and the fact that any two distinct vertices have a path between them is vacuously true. However, Euler's…
Princess Mia
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Prove that connected graph $G$ with $\delta (G)\geq 3$ contains a cycle $C$ such that $G-E(C)$ is still connected.

In a connected simple graph every vertex has a degree at least $3$. Prove that the graph contains a cycle such that the graph remains connected when the edges of this cycle are deleted. Source: https://www.komal.hu/verseny/2000-02/mat.e.shtml I've…
nonuser
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Show that if $G$ is simple a graph with $n$ vertices and the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.

I'm trying to pick up a little graph theory out of Bondy and Murty's Graph Theory as suggested here. Problem 1.1.12 has given me a little hitch. Let $G$ be a simple graph of order $n$ and size $m$. (So there are $n$ vertices and $m$ edges). If…
yunone
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Are these two graphs isomorphic? Why/Why not?

Are these two graphs isomorphic? According to Bruce Schneier: "A graph is a network of lines connecting different points. If two graphs are identical except for the names of the points, they are called isomorphic." Schneier, B.  "Graph…
voices
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Connected components both topologically and graph-theoretically

Given a topological space $(X,\tau)$, a connected component is a maximal connected subset of $X$ with respect to $\tau$ (which exists). Now, consider the group $G$ of homeomorphisms $\varphi:X\to X$; it naturally induces a graph on $X$ where an edge…
Akira Satou
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Does any connected graph G have a spanning tree T with the same domination number?

Let $G$ be a simple graph. A spanning tree of a connected graph $G$ is an acyclic connected subgraph $T$ of $G$ such that $V_T = V_G$. A dominating set of $G$ is a subset $W$ of $V_G$ such that every vertex in $V_G\setminus W$ is adjacent to some…
Esteban Contreras
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Prove a graph is cyclic

I encountered the following problem: Let $G(V, E)$ be a finite undirected graph with $2n$ vertices, in which every connected component has an even number of vertices. Given that exactly $n$ of the vertices of $G$ have a degree at least $3$, prove…
D. Petrov
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3- regular connected graph $G$ has independent subgraph $H$ such that $G-H$ is still connected.

There are $1998$ cities in Dussia, each being connected (in both directions) by flights to three other cities. Any city can be reached by any other city by a sequence of flights. The DSS (Dussia secret service) plans to close off $200$ cities, no…
nonuser
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A problem about the connectivity of vertices that must have the same color for any proper minimal coloring of a graph.

The question is now also published in MathOverflow (here). I'm trying to solve a problem about connectivity of entangled vertices in a graph. Two vertices $u, v$ of a finite graph $G(V, E)$ are said to be entangled if for any proper coloring…
Alma Arjuna
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Inequality relating connectivity, edge connectivity and minimum degree (Whitney's theorem)

The following is a theorem given in Bella Bollobas's Modern graph theory (Springer 2002) Page-73 If $G$ is nontrivial (that is, has at least two vertices), then the parameters $\delta(G)$, $\lambda(G)$ and $\kappa(G)$ satisfy the following…
RagingBull
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What is the probability that the graph remains connected?

Consider the complete graph $K_4$ with four vertices; all vertices are connected by an edge to all other vertices. Suppose now that we flip an unbiased coin for each edge. If heads comes up, we leave the edge where it is, if tails comes up we remove…
cheesewiz
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Finding a spanning tree with at least 100 leaves

I have the following graph theory problem: In a country there are pairs of towns connected by roads in such a way that you can get from any town to any by those roads. The president of the country had ordered to build several new towns and roads so…
Ellie
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Find a graph such that $\kappa(G) < \lambda(G) < \delta(G)$

How would I go about constructing a graph that satisfies this inequality? I am new to graph theory so I'm not sure where to start. Note that: $\kappa(G)$ is the vertex-connectivity of G, the size of the smallest separating set of G. (A separating…
A.B
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A 2-connected graph contains a path passing through all the odd degree vertices

I am trying to prove the above as an exercise in the topic of connectivity. I have tried to do so using ear decompositions, as odd degree vertices may be characterized as end points of ears, but to no avail. Any recommendations are…
ChristianHollis
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Edge-connectivity of edge-transitive graphs

If $G$ is a connected vertex-transitive graph then it is well known that the edge connectivity of $G$, $\kappa'(G)$ equals to $\rm{val}(G)$ - the degree of a vertex in $G$ (note that $G$ is regular.) My question is the following Is $\kappa'(G) =…
Jernej
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