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Questions tagged [graph-theory]

Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.

6
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1answer
186 views

Naive definition of treewidth

Treewidth has arguably pretty involved definition. Recently I was thinking about a problem and turns out it easy to solve it for graphs with small ``naive treewidth''. Naive treewidth is defined as ...
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0answers
51 views

Is there a standard name for this way of modifying graphs?

Let $G = (V, E)$ be an undirected graph. Let me take an edge $\{x, y\}$ (in blue in the drawing) such that $x$ and $y$ have other incident edges. Among the incident edges we choose one edge $e_x = \{...
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0answers
26 views

How to prove Euler formula for graph theory? [closed]

How to prove Euler formula for hypergraph? How we defines faces in hyper graphs ?
0
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1answer
47 views

Counting sum of parities of cycle covers in cubic, planar, bipartite graphs

Let $G$ be a cubic (i.e. every degree exactly three), planar, bipartite graph. By Hall's theorem its edges can be partitioned into three perfect matchings. Take any such partition $M_0,M_1,M_2$ and ...
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0answers
29 views

How to estimate the maximum and minimum eigenvalue of random walk Laplacian graph?

I'm wondering how to estimate the maximum and minimum eigenvalues of random walk Laplacian graph ! The normalized version of graph Laplacian allow to get eigenvalues in range [0,2]. Thank you
3
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0answers
69 views

Generating a random connected bipartite graph

A (n, m, k)-bipartite graph is a bipartite graphs with: independent sets of size $\{n, m\}$ a total of $k \geq n+m-1$ edges We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
5
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0answers
162 views

What's the fastest known algorithm for finding the diameter of a graph?

Given a positively weighted graph what's the fastest algorithm for finding the diameter for that graph?
5
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1answer
141 views

Viola's Reduction of 3XOR to listing triangles

Apparently this was due to Pătraşcu, but in this report on the ECCC server, Viola states that 3XOR can be reduced to listing triangles. Assume that given a graph in adjacency list format, with $m$ ...
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0answers
121 views

Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
1
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1answer
113 views

maximize edges minus vertices in a weighted graph

for a given weighted vertices and edges graph, we want to find the maximum subgraph. the maximum subgraph is made of some vertices and some edges of the given graph which sum of the edges minus sum of ...
6
votes
1answer
122 views

Separating words and graph isomorphism

I wonder if there are any known implications of Babai's recent quasi-polynomial time algorithm for Graph Isomorphism to separating words by DFA's. In both cases the ultimate goal is to differentiate ...
1
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0answers
45 views

Is there an efficient way to reduce a set of graphs under isomorphism?

Suppose we have a set $S$ of (fixed-size) graphs, many of which are isomorphic to each other. How do you find a minimum-size set $M$ of graphs such that every $g\in S$ is isomorphic to some graph in $...
1
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1answer
110 views

Is balanced Hamiltonian cycle NP complete on maximal plane graphs?

I know that the Hamiltonian cycle is NP complete on the class of maximal plane graphs. If we instead ask about balanced Hamiltonian cycles (i.e. same number of faces on both sides) on maximal plane ...
1
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1answer
76 views

Pulling a graph across a partition

I am looking for the name for a particular graph property, if it has been studied, and efficient algorithms for computing it, if they exist. I realise that this may be a well known property that I am ...
1
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1answer
75 views

Cryptography protocols using graph problem instances

I personally am only aware of basic examples of public key cryptography and I haven't studied cryptography yet. I'm curious if there are circumstances in cryptography where using problem instances ...
2
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0answers
119 views

Crime prevention using graph theory and machine learning

I am looking for a way to the model the incidence of crime among a network of individuals. Part of it will use machine learning, and part of it will have to resort to some graph theoretic ...
8
votes
1answer
287 views

Is there an algorithm that finds the forbidden minors?

The Robertson–Seymour theorem says that any minor-closed family $\mathcal G$ of graphs can be characterized by finitely many forbidden minors. Is there an algorithm that for an input $\mathcal G$ ...
9
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0answers
126 views

Random unbalanced bipartite graphs are good small set expanders

My question is about small set expansion properties of random unbalanced bipartite graphs. Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
4
votes
1answer
70 views

Complexity of finding Exact Size Cut-Sets in Bipartite Graphs

I am interested in the problem of deciding if a cut-set of a given size $k$ (i.e. the number of edges crossing the partitions is $k$) exists in a given bipartite graph (both the graph and $k$ are part ...
1
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1answer
44 views

Problem property name where an optimal solution in a graph can be used as a solution in any subgraph

Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
5
votes
1answer
108 views

Counting/Enumerating Minimal Edge Covers

A Minimal Edge Cover is an Edge Cover such that no other Edge Cover is a proper subset of it. Questions Which is the complexity of counting Minimal Edge Covers? Do we know any non-trivial ...
12
votes
1answer
823 views

Number of 4 cycles

Let $C_4$ be a cycle with four vertices. For an arbitrary graph $G$ with $n$ vertices and m edges say $m>n\sqrt n$, how many $C_4$s exist? Is there a lower bound for this?
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0answers
51 views

Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity

I am wondering if this is a topic that has had research done... If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
8
votes
1answer
184 views

Could chromatic number be easy to calculate when colouring is hard for some graph class?

Similar question was asked before, but there was an error in it so it was left unanswered Graph class with easy chromatic number, but NP-hard coloring Is there any infinite set of graphs $C$ such as: ...
4
votes
1answer
119 views

Is graph connectivity definable in existential MSO with vertices and edges?

Can $\exists$MSO$_2$ express graph connectivity? Monadic SO (MSO) is the fragment of second-order logic in which the second-order quantifiers range over relations of arity 1 only. $\exists$MSO is the ...
9
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0answers
98 views

Graphs with minimal-size induced subgraphs

I consider undirected graphs $G = (V, E)$ for which I write $\text{n}(G) := |V|$ the number of vertices and $\text{m}(G) := |E|$ the number of edges. For $d \in \mathbb{N}$, I say that $G$ is $d$-...
4
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0answers
112 views

A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
1
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1answer
76 views

Minimum graph cut with constraints

Given an undirected acyclic raph $G = \{V,E\}$, with each edge $e$ having weight $c_e$in the range $[-\infty, +\infty] $, I want to compute a partition of the graph into $N$ disjoint sets $G_i, i=1,......
6
votes
1answer
197 views

Paths of length $p$ in a Graph, $p$ a prime

I came across the following decision problem, for which I wondered whether anybody of you came across a similar problem and can give me some insight on its nature/complexity. Given as input a ...
-2
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1answer
101 views

Finding Cheapest n-Path [closed]

Given a weighted directed acyclic graph, how can I find the cheapest path from an Origin Vertex to a Destination Vertex which ...
4
votes
1answer
177 views

Fast algorithm to find a maximum connected subgraph of k vertices

Given an undirected graph $G = (V, E)$ and a function $f: 2^V \to \mathbb{R}^+,$ where $2^V$ is the set of all subsets of $V$. Find a connected subgraph $T = (V_T, E_T)$ of k vertices such that $f(V_T)...
6
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0answers
94 views

Ramsey number for path in a colored tournament

I am looking about the following Ramseyish theorem with additional structure on tournament (see [1]). For all number of colors $k$, there exists $n=T(k)$ such that all tournaments of size $n$ ...
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2answers
95 views

Looking for an algorithm to construct a graph from two subgraphs

I am looking for an algorithm to construct a graph from two subgraphs. The problem is as following: Given two graphs g1(V, E) and g2(V, E), find a graph G(V, E) where V(g1) ⊆ V(G), V(g2) ⊆ V(G), E(g1)...
3
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0answers
71 views

Mutually exclusive replacement paths, an existing problem?

A replacement path is a simple path allocated to an edge $e \in G$ that connects the endpoints of $e$ in $G \setminus \{e\}$. The problem An undirected graph $G$ is given and the task is to allocate ...
-1
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1answer
58 views

How to make any graph 2-degenerate?

I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
0
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0answers
48 views

Complexity of extending $P_4 $-partition of cubic graphs

This is a question I posted on MathOverflow before but never got an answer. I am cross-posting it here. Surprising phenomena occurs when we want to extend a partial solution of some easy problems. We ...
4
votes
2answers
142 views

Check if graph stays connected after edge swap

Checking whether a (simple, undirected) graph is connected can be done in linear time in the number of edges. What I am looking for is a more efficient way of checking whether it stays connected after ...
3
votes
1answer
199 views

What exactly is Lawler's modification to Yen's algorithm and how does it work?

I recently read about Yen's algorithm, I understand the algorithm and it seems correct, however Wikipedia mentions that there exists "Lawler's modification" to the algorithm, which is described as ...
4
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1answer
101 views

Probability of a $k$-path in a random graph

Assume that $G\in G(n,p)$; if $p=\frac{\ln n +\ln \ln n +c(n)}{n}$, the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
4
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1answer
379 views

Minimum Union-Sum Cost Path

I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
1
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0answers
42 views

Random Multigraph ER-like model?

I was looking into multigraphs recently and I couldn't find a simple "goto" model for generating random multigraphs along the lines of the ER model of simple graphs. Specifically, I was hoping to find ...
-1
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1answer
51 views

NP-Complete graph problems where a special vertex is given as input?

I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
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0answers
54 views

k-center 2.0: A stronger k-center condition

Given an unweighted, undirected graph, we can use the classical 2-appx for $k$-center to select a set $S$ of centers such that every vertex is within a distance of 2 of some center in $S$. Note that ...
1
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1answer
105 views

Total flow using minimum number of edges on a bipartite network

If I have a set of sources $S$ with total capacity $C$ and a set of sinks $T$ with the same capacity $C$, but not necessarily the same cardinality, is there an efficient way to find the minimum number ...
5
votes
1answer
144 views

Lower bound for triangle-free graphs

I was reading a set of notes where it says It can be shown that $\Omega(n^2)$ space is needed for one-pass algorithms to determine if an (unweighted, undirected) graph $G$ with $n$ nodes contains ...
1
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0answers
39 views

Counting vertex covers on a chain of k nodes that do not contain a sub-chain of length >=3

By a "chain of k nodes", I mean k nodes lined up like a linked-list: o-o-o.....-o . By "do not contain a sub-chain of length >=3", I mean that no cover should contain two edges that shares a node. ...
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0answers
35 views

Planarity testing of directed graph and 3d grid

I wonder if there exist definitions (and known algorithms) of planarity testing for the following case: 1- A directed graph let $G=(V=\{1,\ldots,n\},E)$ be a directed graph. Assume $e_{ij} =(v_i,...
4
votes
1answer
180 views

How hard is it to determine the chromatic number of a unit distance graph?

For example, is it NP-complete to decide whether a unit distance graph is 3-colorable?
5
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2answers
185 views

The chromatic number of a graph as a functor

I was fooling around with some concept and was wondering if this viewpoint is explored at all. Let INJ-GRAPH be the subcategory of graphs (with morphisms as homomorphisms) whose morphisms consist only ...
5
votes
0answers
148 views

Can graphons be useful outside of extremal graph theory?

I was recently introduced to the notion of a Graphon ([Quick introduction], [Original definition]) and I have been fascinated by the idea behind them and the way they combine fields such as graph ...