Questions tagged [graph-theory]

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

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1answer
393 views

Strongly Regular Graph and GI-Completeness

It is not known if graph isomorphism (GI) for strongly regular graphs (SRGs) is in P. Are there any hints that it might or might not be GI-Complete? Are there any strong consequences in such cases? (...
16
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1answer
618 views

What is the fastest deterministic algorithm for dynamic digraph reachability with no edge deletion?

What is the best deterministic result for maintaining the dynamic transitive closure in a directed graph with only edge insertion? I read some papers on the dynamic transitive closure problem with ...
16
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1answer
657 views

Making a minimum-width tree decomposition lean in polynomial time

As is well known, a tree decomposition of a graph $G$ consists of a tree $T$ with an associated bag $T_v \subseteq V(G)$ for each vertex $v \in V(T)$, which satisfies the following conditions: Every ...
16
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1answer
925 views

Number of Hamiltonian cycles on random graphs

We assume that $G\in G(n,p),p=\frac{\ln n +\ln \ln n +c(n)}{n}$. Then the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
16
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1answer
1k views

NP-hardness of a graph partition problem?

I'm interested in this problem: Given an undirected graph $G(E, V)$, Is there a partition of $G$ into graphs $G_1(E_1, V_1)$ and $G_2(E_2, V_2)$ such that $G_1$ and $G_2$ are isomorphic? Here $E$ is ...
16
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1answer
430 views

What is the complexity of this graph problem?

Given a simple undirected graph $G$, find a subset $A\neq \emptyset$ of vertices, such that for any vertex $x\in A$ at least half of the neighbors of $x$ are also in $A$, and the size of $A$ is ...
16
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1answer
540 views

Sensitivity of Graph Properties

In [1], Turan shows that the sensitivity (called "critical complexity" in the paper) of a graph property is strictly greater than $\lfloor {1\over 4} m \rfloor$ where $m$ is the number of vertices in ...
16
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1answer
685 views

Reference for mixed graph acyclicity testing algorithm?

A mixed graph is a graph that may have both directed and undirected edges. Its underlying undirected graph is obtained by forgetting the orientations of the directed edges, and in the other direction ...
16
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0answers
254 views

When does adding edges decrease the cover time of a graph?

When first learning about random walks on a graph $G$, one may have an intuitive feeling that adding edges to $G$ will decrease its cover time $C(G)$. However, this is not the case. The path graph $...
15
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6answers
486 views

Global properties of hereditary classes?

A hereditary class of structures (e.g. graphs) is one that is closed under induced substructures, or equivalently, is closed under vertex removal. Classes of graphs that exclude a minor have nice ...
15
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2answers
850 views

Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?

The $k$-cycle problem is as follows: Instance: An undirected graph $G$ with $n$ vertices and up to $n \choose 2$ edges. Question: Does there exist a (proper) $k$-cycle in $G$? Background: For any ...
15
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5answers
863 views

References for Modular Decomposition

What are good papers/books to better understand the power of Modular Decomposition and its properties? I'm particularly interested in algorithmic aspects of Modular Decomposition. I have heard that ...
15
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3answers
963 views

Subgraph isomorphism with a tree

If we have a large (directed) graph $G$ and a smaller rooted tree $H$, what is the best known complexity for finding subgraphs of $G$ isomorphic to $H$? I am aware of results for subtree isomorphism ...
15
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3answers
2k views

Complexity of edge coloring in planar graphs

3-edge coloring of cubic graphs is $NP$-complete. Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". What is the complexity of 3-edge coloring of cubic ...
15
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1answer
4k views

Partition a graph into node-disjoint cycles

Related problem: Veblen’s Theorem states that "A graph admits a cycle decomposition if and only if it is even". The cycles are edge disjoint, but not necessarily node disjoint. Put another way, "The ...
15
votes
1answer
613 views

Is this dense version of Kruskal's algorithm well-known?

About a year ago, a friend and I thought of a way to implement Kruskal's algorithm for dense graphs in better than the usual $O(m \log m)$ bound (without assuming pre-sorted edges). Specifically, we ...
15
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1answer
782 views

Modular Decomposition and Clique-width

I am trying to understand some concepts about Modular decomposition and Clique-width graphs. In this paper ("On P4-tidy graphs"), there is a proof of how to solve optimization problems like clique-...
15
votes
1answer
839 views

Online transitive closure better than O(N^2) per edge addition

I'm looking for an online algorithm to maintain the transitive closure of a directed acyclic graph with a time complexity less than O(N^2) per edge addition. My current algorithm is like this: ...
15
votes
1answer
484 views

Drawing graphs with few “sharp” vertices?

For a planar embedding of a planar graph on a plane with straight edges, define a vertex as a sharp vertex if the maximum angle between two consecutive edges around it is more than 180. Or in other ...
15
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1answer
725 views

Hardness of Computing Weisfeiler-Lehman labels

The 1-dim Weisfeiler-Lehman algorithm (WL) is commonly known as canonical labeling or color refinement algorithm. It works as follows : The initial coloring $C_0$ is uniform, $C_0(v) = 1$ for all ...
15
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2answers
656 views

GI-hard graph problem not known to be $NP$-complete

Graph Isomorphism ($GI$) is good candidate for $NP$-intermediate problem. $NP$-intermediate problems exist unless $P=NP$. I'm looking for natural problem that is hard for $GI$ under Karp reduction (A ...
15
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1answer
759 views

Decomposing k-connected graphs into (k+1)-connected components

A connected graph can be decomposed into its biconnected components. This block cutpoint tree is unique. Similarly, biconnected graphs can be decomposed into triconnected components. The corresponding ...
15
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1answer
474 views

Imperfect subgraph isomorphism

Consider the following problem: Given a query graph $G = (V, E)$ and a reference graph $G' = (V', E')$, we want to find the injective mapping $f : V \rightarrow V'$ which minimizes the number of edges ...
15
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1answer
439 views

Graph decompositions for combining “local” functions of vertex labelings

Suppose we want to find $$\sum_x \prod_{ij \in E} f(x_i,x_j)$$ or $$\max_x \prod_{ij \in E} f(x_i,x_j)$$ Where max or sum is taken over all labelings of $V$, product is taken over all edges $E$ for a ...
15
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1answer
317 views

What can we prove with infinite graphs that we cannot prove without them?

This is a follow-up question to this one about infinite graphs. Answers and comments to that question list objects and situations which are naturally modeled by infinite graphs. But there are also ...
15
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1answer
488 views

Decomposing graphs of genus one

Planar graphs are $K_{3,3}$-free. Such graphs can be decomposed into tri-connected components, which are known to be either planar or $K_5$ components. Is there such a "nice" decomposition of ...
15
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0answers
216 views

Mixing properties of random walks on graphs

I have a question about this paper (not behind a pay wall) on the Cheeger inequality for graphs. One of the main ideas of the paper is that random walks on graphs can be used to find sets with small ...
14
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4answers
2k views

Counting the number of vertex covers: when is it hard?

Consider the #P-complete problem of counting the number of vertex covers of a given graph $G = (V, E)$. I'd like to know if there is any result showing how the hardness of such problem varies with ...
14
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2answers
2k views

Justification for the Hungarian method (Kuhn-Munkres)

I wrote an implementation of the Kuhn-Munkres algorithm for the minimum-weight bipartite perfect matching problem based on lecture notes I found here and there on the web. It works really well, even ...
14
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2answers
4k views

Optimal algorithm for finding the girth of a sparse graph?

I wonder how to find the girth of a sparse undirected graph. By sparse I mean $|E|=O(|V|)$. By optimum I mean the lowest time complexity. I thought about some modification on Tarjan's algorithm for ...
14
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2answers
2k views

Generalization of the Hungarian algorithm to general undirected graphs?

The Hungarian algorithm is a combinatorial optimization algorithm which solves the maximum weight bipartite matching problem in polynomial time and anticipated the later development of the important ...
14
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2answers
473 views

Approaches to GI inspired by knot problem

GI and Knot Problem both are problem of deciding structural equivalence of mathematical objects. Are there any results establishing connections between them? Nice connections of knot problem to ...
14
votes
1answer
2k views

Is the longest trail problem easier than the longest path problem?

The longest path problem is NP-hard. The (typical?) proof relies on a reduction of the Hamiltonian path problem (which is NP-complete). Note that here the path is taken to be (node-)simple. That is, ...
14
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4answers
710 views

P-complete problems on trees

This question is related to one of my previous questions, NP-hard problems on trees. I am looking for problems that are P-complete on trees.
14
votes
4answers
317 views

Does the infinite graph of diagonals have an infinite component?

Suppose we connect the points of $V = \mathbb{Z}^2$ using the set of undirected edges $E$ such that either $(i, j)$ is connected to $(i + 1, j + 1)$, or $(i + 1, j)$ is connected to $(i, j + 1)$, ...
14
votes
1answer
1k views

Conductance and diameter in regular graphs

Given an undirected, regular graph $G=(V,E)$, what is the relationship between its diameter - defined as the largest distance between two nodes - and its conductance, defined as $$\min_{S \subset V} ~\...
14
votes
2answers
2k views

Number of mincuts of a graph without using Karger's algorithm

We know that Karger's mincut algorithm can be used to prove (in a non-constructive way) that the maximum number of possible mincuts a graph can have is $n \choose 2$. I was wondering if we could ...
14
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2answers
415 views

A graph parameter possibly related to treewidth

I am interested in graphs on $n$ vertices which can be produced via the following process. Start with an arbitrary graph $G$ on $k\le n$ vertices. Label all the vertices in $G$ as unused. Produce a ...
14
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1answer
314 views

Generating Graphs with Trivial Automorphisms

I'm revising some cryptographic model. To show its inadequacy, I've devised a contrived protocol based on graph isomorphism. It is "commonplace" (yet controversial!) to assume the existence of BPP ...
14
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1answer
400 views

How expensive may it be to destroy all long s-t paths in a DAG?

We consider DAGs (directed acyclic graphs) with one source node $s$ and one target node $t$; parallel edges joining the same pair of vertices are allowed. A $k$-cut is a set of edges whose removal ...
14
votes
1answer
419 views

Hitting odd cycles

Is there anything known about the following problem? Does it make sense at all? What is it called? Is it trivially equivalent to some other problem? What is the time-complexity? Given an undirected (...
14
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2answers
544 views

Add a matching to a Hamiltonian path to reduce the max distance between given pairs of vertices

What is the complexity of the following problem? Input: $H$ a Hamiltonian path in $K_n$ $R \subseteq [n]^2$ a subset of pairs of vertices a positive integer $k$ Query: is there a matching $M$ such ...
14
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0answers
550 views

Is it NP-hard to find (the root of) a small decision tree for a monotone boolean function?

Last year I spent some time trying to prove or disprove the following: Conjecture. Consider a graph $G$ and define a 2-DNF formula $\phi$ that contains a term $x \land y$ iff $x\mathrel{-\!-}y$ is ...
14
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0answers
1k views

What is the fastest deterministic algorithm for incremental DAG reachability?

As the title. The incremental algorithm maintains the reachability information of a DAG when it undergoes a series of edge insertions (but no deletions). And the algorithm supports constant query (if ...
14
votes
0answers
347 views

Finding all-pairs anti-distance

Thanks for a great forum. This is my first post here. I am working on a signal processing application and the core of one the main algorithms reduces to a graph theoretical problem. Let $G=(V,E)$ ...
14
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0answers
412 views

Question on Products of Graphs

Let $S_{n}(G)$, $C_{n}(G)$, $T_{n}(G)$ be the $n$-fold Strong Product, Cartesian Product and Tensor Product of a graph $G$ on $|V|$ vertices. Let the chromatic number ($\chi(G)$) and the independence ...
14
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0answers
484 views

Bi-partite expander graphs

My question relates to bi-partite expander graphs, defined as bi-partite graphs on $n$ left vertices, $m$ right vertices, constant left-degree $k$, such that For any linear-sized subset $S$ of the ...
14
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0answers
499 views

Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)

Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same. First, I will define a few ...
14
votes
1answer
337 views

Space-approximation Trade-off

In their paper Approximate Distance Oracles, Thorup and Zwick showed that for any weighted undirected graph, it is possible to construct a data structure of size $O(k n^{1+1/k})$ that can return a $(...
13
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4answers
3k views

LP relaxation of independent set

I've tried the following LP relaxation of maximum independent set $$\max \sum_i x_i$$ $$\text{s.t.}\ x_i+x_j\le 1\ \forall (i,j)\in E$$ $$x_i\ge 0$$ I get $1/2$ for every variable for every cubic ...

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