Theoretical Computer Science

Questions tagged [graph-theory]

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

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1answer
109 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
445 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 ...
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0answers
53 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 ...
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1answer
53 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
64 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
107 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
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1answer
166 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 ...
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0answers
63 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
36 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
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1answer
189 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
210 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
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0answers
153 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 ...
6
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1answer
141 views

Partition edges into edge disjoint walks

Consider an undirected graph $G=(V,E)$ and two sequences of $k$ vertices $S=s_1,\ldots,s_k$ and $T=t_1,\ldots,t_k$. A set of $k$ walks is called a $(S,T)$-walk partition if the walks form a ...
3
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0answers
122 views

Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$? [closed]

We got an argument that 3-coloring bounded degree graphs is subexponential with complexity $O(\exp{(\sqrt{n}\log^2{n})})$. The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$ and 3-...
4
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1answer
186 views

Can $L=SL$ be shown with the replacement product instead of the zig-zag product?

(This is a bit of follow-up to https://cstheory.stackexchange.com/posts/comments/93266 but is a distinct enough question I though it should be on its own.) In Omer Reingold's logspace USTCON ...
5
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1answer
131 views

Path in a graph with durations [closed]

I have the following problem: given a directed graph $G=(V,E,d)$, where $d:V\to\mathcal{I}(\mathbb{Q}_0^+\cup\{+\infty\})$ (here $\mathbb{Q}_0^+$ denotes the set of non-negative rationals and $\...
3
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0answers
82 views

PTIME or NP-Hardness of stochastic objective function

I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
3
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0answers
90 views

Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
3
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1answer
113 views

Results/concepts that also proved useful outside of their “home areas”

There are some results/concepts in TCS which are used in areas other than the "home area" where they emerged. For example, NP-completeness has complexity theory as its home area, but it is also used ...
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2answers
284 views

Is perfect matching for bipartite graph with no cycles unique?

Given a balanced bipartite graph that satisfies Hall's theorem (is non singular) then it shown that it has at least one perfect matching. My question is if the balanced bipartite graph is also ...
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0answers
59 views

Generating random labelled trees

I am looking for a simple rejection-free algorithm to uniformly sample random labelled trees (i.e. to generate each of them with the same probability). One possibility is to generate Prüfer sequences ...
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0answers
85 views

Is NP-complete the existence of paths of a given length in a directed graph? [closed]

Given a directed graph G= (V,E), a pair of vertices s and t, a natural number K encoded in binary, whether the problem to decide there exists a path (not necessarily simple) from s to t of length K is ...
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1answer
547 views

Closeness Centrality for Weighted Graphs

In order to determine the Closeness Centrality for a vertex u in a graph, you compute the shortest path between u and all other vertices in the graph. The centrality is then given by: $C(u) = \frac{1}...
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0answers
59 views

Finding a Minimal k-Subgraph

Given a complete, positively weighted, bidirectional graph with $n$ nodes without self-loops. Hence the corresponding adjacency matrix $A$ is positive, symmetric, and has zero main diagonal. I am ...
0
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1answer
133 views

Is there any better than (2/k)-approximation algorithm for Independent Set in Coloring graph?

I would like to know any results in terms of approximation algorithm about Maximum weight Independent Set problem in coloring graph. Input: Given a graph G with non-negative vertex weights and ...
5
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79 views

Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
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193 views

Hardness of Approximation for minimum path cover in an undirected graph?

Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path ...
1
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1answer
107 views

How good of an approximate 2-coloring can you get of the halved cube graph?

We say that a 2-coloring $col : V_G \rightarrow \{0, 1\}$ of a graph $G$ is $\epsilon$-approximate if $Pr_{(w, v) \in E_G}(col(w) \neq col(v)) \geq \epsilon$. For every $n$, what is the maximum $\...
2
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1answer
98 views

What is the name of a graph with local clustering coefficients equal to zero?

I am struggling to find in literature a name for a kind of graph where all local clustering coefficients are equal to zero (or, at least, bounded). For instance, domino, a subset of cacti and all ...
10
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2answers
220 views

Common insights into hypothetical complexity of graph problems

I came across two examples of hypothetical hardness of some graph problems. Hypothetical hardness means that refuting some conjecture would imply the NP-completeness of the respective graph problem. ...
2
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1answer
111 views

Is Eulerian Path (or Eulerian Cycle) definable in Monadic Second Order Logic?

Does there exist a monadic second order logic formula which is satisfied by a graph if and only if it has an Eulerian path (or Eulerian cycle). I am looking for properties of graphs which are ...
16
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1answer
420 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 ...
3
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0answers
525 views

Eigenvalues of adjacency matrix of a connected bipartite graph

Let $G=(V,E)$ is a connected d-regular bipartite graph with the same number of vertices on both sides of the bipartition. It's known that that the largest eigenvalue of its adjacency matrix would be d,...
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1answer
221 views

Does any DAG can be topologically sorted? [closed]

I am not good enough in computer science. My intention is to solve some programming problem in terms of DAG's. The key point is that before getting them into database, I need run "topological sort" in ...
3
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1answer
174 views

Natural (well studied) classes of graphs with treewidth $\Theta(n^\alpha)$ with $1/2 < \alpha < 1$

Which natural (well studied) classes of graphs have treewidth that scales as $\Theta(n^\alpha)$ in the number $n$ of vertices, with $1/2 < \alpha < 1$?
0
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1answer
48 views

Maximize graph with k cut edge operations

I have undirected graph with N nodes each with some weight. There are M edges and in exactly K operations I want to maximize the XOR sum of connected components of the graph. ((n1 XOR n2 XOR n3) + (c1 ...
4
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0answers
127 views

Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
2
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0answers
115 views

Practical polynomial-time implementation of bounded degree graph isomorphishm

There's a well-known article for solving graph isomorphism problem in polynomial time. Many other articles on the subject of isomorphism mention it as a possible "alternative", but note that is not ...
5
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2answers
243 views

Efficient way to generate random planar cubic bipartite graphs

3-regular bipartite planar graphs appear in a variety of NP- / #P-complete problems. Suppose one wants to test the complexity of these problems via numerical experiments. Is there an efficient way to ...
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2answers
93 views

reference for a special modular decomposition

For modular decomposition. X is a module if all members of X have the same set of neighbors among vertices not in X. I need a special modular decomposition. X is my module if all members of X have ...
2
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1answer
170 views

Will core decomposition get a maximal clique?

I have read David Eppstein's paper about maximal clique enumeration by using degeneracy order. It has mentioned the core decomposition, which is removing the vertex with the smallest degree ...
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0answers
26 views

Complexity of recognizing generalized graph join

A join of two graphs is the union of both graphs with additional edges such that every vertex of the first graph is connected to every vertex of the second graph. There is a generalization of this, ...
2
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1answer
67 views

Are equally weighted MSTs closely related?

Suppose we have an undirected connected graph $G=(V,E)$ that has several minimum spanning trees. We say two trees $T_1, T_2$ are connected if they share exactly $|V|-2$ edges(*). In other words $T_1$ ...
2
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0answers
132 views

Maximum weight non-overlapping paths in a DAG

Suppose we have a weighted DAG $G$. A $m$-path-tuple is defined as $(P_1, ..., P_m)$ in which $P_i$ is a path on the graph, and no $P_i$ and $P_j$ share any edges. In other words each edge of the ...
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0answers
153 views

Minimum cut on a directed graph with negative term

Suppose we are given a directed Graph $G=(V,E)$ and there is a nonnegative weight $w(u,v)$ is defined for the edge from $u$ to $v$. The task is to partition vertices to $A$ and $B$ (partition means $V ...
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0answers
100 views

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_{...
2
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0answers
94 views

Unbalanced connected partition

Let $G = (V, E)$ be a connected graph with (possibly negative) vertex weights $w(v)\in\mathbb{Z}$. We want to partition the vertices into two parts such that the induced graphs $G'$ and $G''$ are ...
7
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1answer
432 views

Why isn't the Charikar algorithm for finding the densest subgraph optimal?

I read about the algorithm in Greedy Approximation Algorithms for Finding Dense Components in a Graph by Moses Charikar, and I tried to find an instance/graph where the algorithm returns a different ...
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1answer
123 views

Is a k-connected graph also k-1 connected?

So I'm reading about k-connected graphs, and I'm a little confused about them. This is the main definition I've seen: A graph is k connected if and only if for any distinct x, y vertices in the graph,...
7
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1answer
135 views

Random Deterministic Automata

I am familiar with the term of random graphs, such as $G(n,p)$- a distribution over simple undirected graphs with $n$ vertices, where each edge appears in a graph w.p. $p$. That is, each graph $G=(V,E)...

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