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Questions tagged [graph-theory]

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

5
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0answers
91 views

Graceful labeling completion problems

A graceful labeling of a graph with $m$ edges is a labeling of its vertices with some subset of the integers between $0$ and $m$ inclusive, such that no two vertices share a label, and such that each ...
8
votes
1answer
146 views

Decomposition of edges of eulerian graph into maximum number of cycles

I'm interested in the following problem. Given an eulerian graph $G=(V,E)$, we are to find a partition of its edges $C_1, C_2, \ldots, C_k$ ($\cup_i C_i=E$ and $i \neq j \leftrightarrow C_i \cap C_j =...
1
vote
1answer
82 views

Complexity of listing all minimal cut sets / connected 2-partitions of a graph

I am trying to find an algorithm that would give me for a given graph all minimal cut sets or equivalently all ways to partition the graph in two connected components. I am searching for an algorithm ...
4
votes
0answers
488 views

Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
2
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0answers
53 views

Complexity consequence of logarithmic boolean width of co-bounded degree graphs?

The paper On graph classes with logarithmic boolean-width claims that the boolean width of co-k-degenerate graphs is at most $k\log{n}$ and a lot of graph vertex partition problems can be solved in ...
6
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0answers
123 views

Book/ Monograph on graph minor theory [Reference request]

I want to learn graph minor theory. Now i have read the very basic things and the overview from the book of R.Diestel but proceeding further is getting difficult. Currently, I am also following the ...
2
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0answers
53 views

The algebraic connectivity of graphs with large isoperimetric number

I asked this question on MO, but no answer. Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$. the isoperimetric number of $G$, denoted $i(G)$, is defined by $$i(G) = \min_{|S| \leq |...
6
votes
1answer
122 views

Directed graph with bounded in-deg can be partitioned in a balanced way

I want to prove that for all $n$, there exists a constant $c(n)$ such that if $G=(V,E)$ is a directed graph with in-degree bounded by $n$, it is possible to partition the set of vertices $V$ into two ...
3
votes
1answer
758 views

Minimal Cost of Eulerian Path

Problem: Given a planar (undirected and mostly sparse) graph with an Eulerian Path, we introduce a cost function f: (v, e1, e2) for all two edges e1 and e2 that share a vertex v. The function also ...
2
votes
1answer
64 views

Almost regular subhypergraph of hypergraph with large minimal degree

I am interested in knowing whether the following conjecture is true or not: For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$. ...
5
votes
1answer
273 views

Modifying Edmonds' Blossom Algorithm

Given a connected road network on an Island without one-way streets, where should I para-shoot in and what route should I take to deliver mail to all houses on the island (being picked up again by ...
1
vote
1answer
420 views

Clique cover problem

Consider the following graph problem. We are given a graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. For each vertex $...
0
votes
1answer
95 views

What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?

We know counting perfect matching for bipartite graphs with vertex degree $2$ is in $P$ while counting perfect matching for graphs with vertex degree $3$ is in $\#P$. We also know there are degree $3$...
3
votes
1answer
172 views

Existence of $d$-regular expander graph that can be represented as a bipartite graph

It is easy to derive from the definition of expander graphs that a $n$-vertex expander graph $G$ does not have a $o(n)$-vertex/edge separator. I was wondering if we can build a $d$-regular expander ...
0
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0answers
47 views

well-behaved outerplanar

I found a cite [1] here for a graph that use the term "Well behaved outerplanar graph" But it does not have a explicit definition. Does it mean that is a related family of graphs that a bit different ...
5
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0answers
122 views

reference clarification: Whitney's theorem on unique embeddability of 3-connected planar graphs?

This is a question about the correct reference for a result that seems to appear frequently in the literature on planar graph isomorphism. In "A $V \log V$ Algorithm for Isomorphism of Triconnected ...
2
votes
1answer
90 views

Counting xyz-graphs in $\mathbb{Z}_n^3$

This is a followup question to: Lower bound on the largest restrained cubic subset How many distinct xyz-graphs exist in $\mathbb{Z}_n^3$? We denote this number as $C(n)$ This question may be seen ...
3
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2answers
122 views

Lower bound on the largest restrained cubic subset

Consider an $n \times n \times n$ cube. I would like to consider subsets of points in the cube with the two following constraints: Each row in the cube (in any of the three directions) has exactly 2 ...
6
votes
1answer
165 views

Graph rewriting with one-to-many pattern matching?

In the single-pushout approach to graph rewriting, many nodes in a pattern graph can be matched to a single node of a target graph. My question is if there is a notion of graph rewriting where the ...
-1
votes
1answer
131 views

A conceptual question regarding hardness proofs by reduction [closed]

If we restrict the input domain of a known NP-hard problem P so that this restricted domain is equal to the input domain of another problem S, then show that we can reduce a solution to P given input ...
2
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0answers
79 views

Graphs for which the number of shortest paths between every pair of vertices is polynomially bounded

Let $G$ be a graph with $n$ vertices and $m$ edges, such that for every two vertices $u$ and $v$, the number of shortest paths from $u$ to $v$ is bounded by some polynomial $poly(n,m)$ in $n$ and $m$. ...
4
votes
2answers
394 views

Is this vertex ordering optimization NP-Hard?

Could you help me to prove that the following problem is NP-hard? Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
0
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0answers
127 views

How to partition a graph while minimizing the count of intra-edges?

Suppose we are given an undirected graph $G(V,E)$. The objective is to partition the graph vertices $V$ into $n$ partitions $P_1, P_2, ..P_n$ such that following criteria is satisfied: $|P_i| = |V|/n$...
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0answers
71 views

limit the total flow for some edges in an directed graph

How can I limit the total flow for some edges in an directed graph. We know that every edge can have an upper and lower bound solely. But apart from this sole upper and lower bounds for each edge, if ...
4
votes
1answer
114 views

Hadwiger number under matching contraction

Given a graph $G$ with Hadwiger number $h(G)$ and a matching $M$ of $G$. Let $G/M$ be the simple graph obtained by contracting $M$. I am looking for a lower bound on the Hadwiger number of $G/M$ as a ...
0
votes
1answer
97 views

Reachability in Dynamic Line Graph

Given a directed line graph $G = v_1 \rightarrow v_2 \rightarrow \cdots \rightarrow v_n$, there are two operators, namely $\mathsf{move}(v_i, v_j)$: this operator moves $v_i$ to the position ...
3
votes
2answers
74 views

Maximal non-reducible vertex cover of a graph

Let $G=(V,E)$ be a graph (i.e. an undirected simple finite graph). We say that a vertex cover $V'$ of $G$ is non-reducible if any $V''$ with $V''\subsetneq V'$ is not a vertex cover of $G$. We say ...
0
votes
0answers
92 views

Path-width and chromatic number

How would I prove that the chromatic number of a graph $G$ is smaller than or equal to the path-width of $G$ + 1? or $\chi(G) \leq pw(G) + 1$
-1
votes
1answer
143 views

Is there a name for graph regions that lie between two nodes?

Perhaps a bit more formally, is there a name for regions delimited by nodes A and B, in a directed graph, where all paths starting from A, when prolonged, will eventually reach B, and all paths ...
0
votes
1answer
53 views

How to check whether graph of n vertex contains n/k disjoint k - complete graph by linear programming? [closed]

Edges are given in form of $X_{ij}$, which denotes whether there is edge in between $i^{th}$ and $j^{th}$ vertex. I am solving integer optimization problem and want to add this constraint to it.
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0answers
212 views

Can Lexicographic BFS be implemented in logspace?

Input: Given graph $G=(V,E)$ vertex labeling in some order Output: Change the labeling of vertices's such that labeling start $v_1$ as $u_1$, next label the neighbors of $v_1$ as $u_2,u_3,u_4,...$ ...
7
votes
2answers
237 views

What is the connection between moments of Gaussians and perfect matchings of graphs?

Today, I heard the following statement in a talk: The 4th moment of a $1$-dimensional Gaussian distribution with mean $0$ and variance $1$ is the same as the number of perfect matchings of a ...
7
votes
1answer
156 views

Automata : Language Containment, Minimality & Graph Homomorphism

Given two DFAs $A$ and $B$ defined on the same alphabet, a (graph) homomorphism $h:A \rightarrow B$ from $A$ to $B$ is a mapping of the states of $A$ into the states of $B$ such that : if the state $...
6
votes
2answers
231 views

Log space algorithms for modular decomposition tree

Can we have log space algorithms for modular decomposition tree (see definition) for any graph? If not, can we have log space algorithms for modular decomposition tree for any particular graph class? ...
6
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0answers
96 views

Optimal polynomial time algorithm to determine if a random graph is $k$-colorable

Let $G(n, d/n)$ be an Erdos-Renyi graph with edge probablity $p = d/n$. For any fixed $k$ sufficiently large, it is known that $d_{k-col} \sim 2 k\log k$ is the sharp threshold for $G(n, d/n)$ to be $...
9
votes
1answer
215 views

Vertex isoperimetric number of a graph - NP-hard?

The vertex isoperimetric number of a graph $G=(V,E)$ is $i_V(G) = \min\{\frac{|N(S)|}{|S|} : S \subseteq V, 1\le |S|\le \frac{|V|}{2}\}$. Several academic papers state that the problem of computing ...
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0answers
83 views

Non-backtracking paths and the zeta function of graphs

This question has also been posted on mathSE here: https://math.stackexchange.com/questions/2215888/non-backtracking-paths-and-the-ihara-zeta-function For a connected $d$-regular graph $G=(V,E)$ with ...
4
votes
2answers
385 views

A generalization of edge cover

Suppose we are given a general (connected) undirected graph $G = (V, E)$. An EDGE COVER asks a set $S\subseteq E$ of the minimum number of edges, such that each vertex $v\in V$ is incident to at least ...
15
votes
2answers
514 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 ...
0
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0answers
75 views

Looking for a list of algorithms that are more efficient for an outerplanar graph than for an arbitrary graph

I'm Looking for a list of algorithms that are more efficient for an outerplanar graph than for an arbitrary graph. I found only the following: ...
8
votes
0answers
212 views

Embedding a graph with specified edge lengths in d-dimensional space

Given any undirected edge-weighted graph (with weights > 0) and some dimension d, is there a way to assign positions in $\mathbb{R}^d$ to those vertices such that all of the edges between them have ...
1
vote
1answer
117 views

Number of solutions to the biclique cover problem

Given a bipartite graph and the number bicliques K, how many ways exist to solve the biclique cover problem using K (possibly empty) bicliques?
13
votes
1answer
514 views

Are there interesting graph classes where the treewidth is hard (easy) to compute?

Treewith is an important graph parameter that indicates how close a graph is from being a tree (although not in a strict topological sense). It is well known that computing the treewidth is NP-hard. ...
0
votes
1answer
420 views

What is “Synthetic Network” in Network Science?

I checked out several articles, books available, but didn't find what exactly is a "synthetic network".
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0answers
113 views

Computing mTSP in a k-complete weighted graph

Short Story: I have a weighted digraph and I need to perform a multi TSP on it. The digraph that i'm working on have the following restrictions: It is a complete digraph, so for every node p and q ...
2
votes
0answers
81 views

What can i learn about a graph about which only certain properties are known [closed]

1) Suppose we are given the following facts about a graph. What can we conclude/compute beyond these facts? The fact that graph $G(V,E)$ is planar, and thus that it is 4-colorable, The degree of each ...
6
votes
1answer
280 views

Connectivity of a random regular graph of degree $d$

An Erdos-Renyi graph over $n$ vertices and average degree $d$ is not connected whp iff $d < \log n$. I was wondering for what the degree $d$ would a random regular graph of degree $d$ be connected? ...
2
votes
1answer
146 views

Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?

The classical representations of the graph automorphism problem is Karp-reducible to the classical representation of the graph isomorphism problem. The sketch of proof for this can be written as ...
2
votes
0answers
146 views

Computational Complexity of cycle double cover

Let $\mathcal{G}$ be the set of all finite simple graphs. Let graph $G\in \mathcal{G}$ and $C_G=\left <C_1,...,C_m \right >$ be a sequence of cycles of $G$ for some $m$. For every edge $e$ of $G$...
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votes
1answer
88 views

Multiple source shortest path with one reversal [closed]

Lets say we have a directed graph G, with vertices V, that have lengths l. I need to find the shortest path between every ordered pair of vertices in the graph, with the following constraint: In a ...