# Questions tagged [graph-theory]

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

1,231 questions
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### 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 ...
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### 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 $3$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $m$. ...
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### 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 $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\chi(G) \leq pw(G) + 1$
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### 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 ...
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### 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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### 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 $u_2,u_3,u_4,...$ ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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?
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### 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. ...
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### 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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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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 $G$...