Questions tagged [graph-theory]
Graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects.
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Given several $a_i$-$r$ paths in a planar graph how ``balanced" of a tree rooted at $r$ can I make?
Suppose I am given distinct nodes $a_1,a_2,.., a_l, r$ and several $a_i$-$r$ paths $P_i$ in a planar graph $G$.
I wish to construct a tree $T$ connecting $a_1,a_2,.., a_l, r$ that minimizes the ...
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Asking boolean question on the nodes of a DAG to find the target node
We are given a DAG $G=(V, E)$ and an unknown target node $x \in V$ to find. There is a mechanism to probe a node, $y \in V$, to ask question of the form "Given the node $y$, is the target node $x$...
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Cover all triangles of a graph with n subgraphs as small as possible
What is the smallest number $s(n,\Delta)$ such that for any undirected simple graph $G=(V,E)$ with $n$ vertices and $\Delta$ triangles, there exist $n$ subgraphs of $G$ covering all triangles where ...
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Finding the shortest cycle containing a vertex in a graph
Given a connected undirected graph with edges $E$ and verticies $V$ and a vertex $v \in V$, find the length of the shortest cycle containing $v$. The best I could do is $O(|E|*deg(v))$, by trying to ...
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(Where) in the polynomial hierarchy is determining the mixing time of an implicitly defined graph?
Consider an implicitly defined graph; for example, let $G$ be a finite group generated with $n$ generators as $\langle g_1,g_2,\ldots g_n\rangle$ and let $\Gamma$ be the Cayley graph of $G$ under ...
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Let $C$ be collection of subsets of $[N]$. Given, $n\in [N]$, what's the terminology for $card\{s\subseteq [N]\mid n\not \in s,\, s \cup\{n\}\in C\}$?
Let $N \ge 1$ be an integer and let $\mathfrak S$ be a nonempty collection of subsets of $[N] := \{1,2,\ldots,N\}$. For any $n \in [N]$, define $\partial_n \mathfrak S := \{S\setminus\{n\} \mid S \in \...
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notation in graph theory [closed]
I was reading a paper, and I found a notation that I don't understand:
$\mathbb{E}[| \textbf{S} |] $, where $\textbf{S}$ is a set. Are there any differences with the notation $\mathbb{E}[formula]$ (I ...
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Cycle packing with degree condition
Given a directed graph where each vertex has the same in-degree as out-degree, I would like to find the maximum number of edge-disjoint cycles. Is this NP-hard?
Without the degree condition, the ...
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What's the connection between branchwidth and treewidth
I understand that treewidth and branchwidth are essentially equivalent for a fixed graph, given that $branchwidth(G) = Θ(treewidth(G))$.
However, my question pertains to a specific case involving ...
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Ensuring the connectivity of an undirected graph through linear programming
I am trying to solve a linear programming problem that deals with finding an optimal subgraph as a function of several parameters. The case is I am trying to model a constraint that ensures that the ...
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END OF THE LINE problem finding a node with in-degree $0$ or out-degree $0$ depending on the initial node
The END OF THE LINE problem is stated as
Given two circuits, P and N, a node, v, is balanced if $P(N(v)) = N(P(v)) = v$ or $P(N(v)) \neq N(P(v)) \neq v$. Given that $0^n$ is not balanced, find ...
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Embedding degree-3 planar graphs as topological minors in wall graphs in polynomial time
For a proof, I need the fact that we can efficiently embed an input planar graph into a representative of a specific family of high-treewidth graphs. Specifically, I need an embedding as a topological ...
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Entries of the Inverse Laplacian
Let $L$ be a graph Laplacian. What is the meaning of the entries of its (pseudo)inverse $L^{-1}$? In other words, are there any interpretations which might help with understanding the entries of $L^{-...
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Cover a graph with complete graphs
I want to find the smallest possible function $k(n,m)$ such that for any graph $G$ with $n$ vertices and $m$ edges, there exists $n$ vertex sets $S_1,S_2,...,S_n\subseteq V$ each with size $k(n,m)$ ...
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Minimum vertex-separators under edge addition
I am trying to prove the following claim.
Let $T$ be a minimum $st$-separator in an undirected graph $G$, and let $x \in T$.
Let $S\neq T$ be a minimal $st$-separator (i.e., not necessarily minimum), ...
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Complexity of induced Steiner Tree problem
Consider the following problem: we are given an undirected graph $G=(V,E)$ and three terminal vertices $t_1,t_2,t_3\in V$. We are asked whether there exists a set of vertices $S\subseteq V$ such that ...
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Two graphs indistinguishable by 4-WL
There are pairs of non-isomorphic graphs that are indistinguishable by the k-WL (but distinguishable by (k+1)-WL) [1]. For example 4x4 rook’s graph and the Shrikhande graph are non-isomorphic but the ...
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increasing minimum graph degree by adding edges
My problem: Given a graph $G=(V, E)$ and an integer $\ell$,add a minimum number of edges to $G$ so that in the resulting graph every vertex has degree at least $\ell$.
Is there a polynomial-time ...
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Treewidth for hypergraphs that specify connectedness requirements
This question is about an alternative definition of treewidth, called weak treewidth. It is defined on hypergraphs where hyperedges intuitively require that the connected subtrees of occurrences of ...
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Maintaining a $K_{3,3}$-minor-free graph
Suppose we are given that an undirected, connected graph $G$ is $K_{3,3}$-minor-free.
Let $a,b\in V(G)$ be non-adjacent vertices.
Under what conditions is the graph that results by adding the edge $(a,...
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Spectral sparsification of graphs with negative edge weights
I am reading the following well-known paper on spectral sparsification of weighted graphs: https://arxiv.org/pdf/0808.4134.pdf. Page 2 contains most of the definitions relevant to this question.
It is ...
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Hardness of Maximum Independent Set in 3-Colorable Graphs
Let $G = (V,E)$ be an undirected graph such that there is a proper coloring of the vertices of $G$ in three colors.
Question: In such graphs, are there known results for the hardness of finding a ...
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Graph partitioning to minimize sum of intra-partition edge weights
I've seen a lot of graph partitioning algorithms w/ the objective of minimizing the weight of inter-partition edges, (e.g. k-way partitioning) but haven't quite found anything on minimizing the total ...
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What is this graph problem, and how hard is it?
My problem is quite simple to state, so it surely must have a name:
Given a graph $G=(V,E)$ with edge weights $w(e) \in \mathbb{Z}$, find a $V' \subseteq V$ that maximizes $\sum_{e \in E' } w(e)$, ...
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On the borderline between natural and artificial problems
While there is no formal definition of what constitutes a natural algorithmic problem,
in most cases there is pretty good consensus whether a specific problem is natural or artificial. Natural usually ...
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Connected dominating set in bipartite graphs
Let $G$ a bipartite graph with two disjoint set of vertices $\mathbf{A}$ and $\mathbf{B}$.
Denote $n_a:=|\mathbf{A}|$, $n_b:=|\mathbf{B}|$. Suppose the following conditions hold:
$\Theta(1)<n_b<...
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Number of vertices that a connected dominating set can reach in densely connected graphs
Consider a undirected densely connected (every vertex has $>\Theta(1)$ incident edges) graph $G$. Denote its vertices set as $\mathbf{V}$, number of vertices as $n$.
A connected dominating set $\...
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Can input-output matrices optimize bidirectional search?
Given a bidirectional search on a weigthed digraph, could a modified input-output matrix guess what nodes are more likely to belong to the shortest path and the search be done through these nodes ...
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2xn grid graphs from ring graphs via local complementations
(Local complementation) A local complementation $\tau_v$ is a graph operation specified by a vertex $v$, taking a graph $G$ to $\tau_v(G)$ by replacing the induced subgraph on the neighborhood of $v$, ...
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Spanning Tree that Preserves the Number of Branch Vertices
Suppose a undirected connected graph $G$, denote the number of vertices in $G$ as $n$, number of branch vertices (i.e., vertices with degree $\geq 3$) as $n_{\geq 3}$. Suppose $n_{\geq 3}>\log(n)$.
...
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Is there a version of Klein-Plotkin-Rao (KPR) Theorem that yields components of small diameter rather than weak diameter?
The Klein-Plotkin-Rao (KPR) Theorem says we can find either a $K_{r,r}$ minor or an
edge-cut of size $O(|E|r/\delta)$ whose removal yields components of weak diameter $O(r^2 \delta)$, that is, any two ...
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Finding a "typical" path
Consider an undirected graph with two distinguished nodes $u\neq v$. How hard is it to find an $u-v$ path, such that its length is as close to the average $u-v$ path length as possible?
Formally, for ...
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Solution for a bipartite demand and supply graph
Given a set of distinct nodes ($A \cup B$) one set represents nodes with a supply ($supply(a), a \in A$) and the other represents nodes with a demand ($supply(b), b \in B$). In a bipartite graph I am ...
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Small set expansion and expanders
Given a graph $G=(V,E)$ on $n$ vertices and $0 \leq \delta \leq 1/2$, we can define the expansion of $G$ over small sets:
$$
h(G,\delta)= \min_{\vert S\vert \leq \delta n } \phi(S) \ ,
$$
with
$$\phi(...
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Packing k vertex trees
Consider a graph $G=(V,E)$ with $n$ vertices.
What do we know about packing of $k$ vertex trees, Both integral and fractional packing are interesting.
$k=2$, it is just the number of edges, hence ...
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Dual of cut of embedded graph disconnects surface
Let $G$ be a graph that embedded on a surface of genus $g$, moreover the embedding is triangulated. Let $C$ be a collection of edges that forms a minimal edge cut for $G$. Let $C^*$ consist of the ...
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Approximative counting of matchings in a graph
The work by Jerrum & Sinclair (1989) describes an approximative approach to determining the number of matchings $|M_\ast(G)|$ in a graph $G=(V,E)$. The fundamental ingredient of the approximation ...
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Unweighted bipartite $b$-Matching
Consider the following problem, of which I am pretty certain that it is polynomially solvable.
Given some arbitrary bipartite Graph $G=(L\cup R,E)$ and some vector $b\in\mathbb{N}^{|L|}$ with $\sum_{i=...
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Are there good analogues to Sparsest Cut/Balanced cut for vertex separators instead of edges cuts?
Most problems about cutting graphs into roughly equal parts such as Sparsest cut, Graph partition, Balanced Cut, etc are based on minimizing the size of an edge cut. Even if all of those problems are ...
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Regularity Lemma for Multi-Relational Graphs?
Is there an analogous to Szemerédi regularity lemma in the setting, where I have multi relational graph i.e. I have $n$ nodes, but instead of having edges to be in $\{0,1\}$ i.e. there is an edge or ...
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Finding Hamilton cycles in random graphs
For a random graph $G$ of minimum degree 3, can we find a Hamilton cycle in linear time (with high probability for every edge density)?
If this is an open problem, I will also accept an empirically ...
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Find the minimum cost spider joining a root to some leaves
A spider is a tree with at most one vertex of degree greater than 2. This vertex is called the head of the spider.
I am interested in the following problem: We are given an undirected graph $G = (V,E)$...
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How to maximize flow through a graph based on edge orientation (in 3D Cartesian Coordinate Space)?
Problem Stmt:
Suppose you have a graph $G$ with edges $E$ and nodes $V$. The nodes have ${x,y,z}$ coordinates in 3D Cartesian space. Assuming each node contains an $x$ amount of material, the idea is ...
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Origin of Berge's (Weak) Perfect Graph Conjecture
In an account of his thought process (refer p. 3) leading up to the perfect graph conjecture (which I'm preparing a seminar talk on), C. Berge states what seems to be a crucial step:
(1) a graph $G$ ...
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What's the difference between 'theoretical' and 'applied' runtime complexity?
I recently followed a course on complexity theory and according to the professor I got one particular question wrong. The question involved the runtime complexity of checking the correctness of a ...
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The "branch-depth" parameter and its use in FPT algorithms
Let $P=(v_1,\dots,v_q)$ be an induced path in the undirected graph $G(V,E)$. In [1], the authors define the branch depth of $P$ to be $b(P)=|N_G[\{v_1,\dots,v_{q-1}\}]|-1$. Further, in [1] it is shown ...
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The tree augmentation problem, but with hyperlinks
In the (Weighted) Tree Augmentation Problem, we are given a tree $T = (V,E)$ and a set of additional edges $L$ called links with non-negative costs. Each link $\ell = (u,v)$ covers the tree edges ...
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Is there FPT or XP algorithms known for Shortest Steiner cycle and $(a,b)$-Steiner path problem
Shortest Steiner cycle and $(a,b)$-Steiner path problem are generalizations of optimization versions of Hamiltonian cycle and Hamiltonian path problems.
The Shortest Steiner cycle problem is defined ...
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Is there an FPT or XP algorithm known for this version of $k$-edge disjoint paths problem?
The shortest $k$-edge disjoint paths problem is defined as follows:
Input: An undirected graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$.
Question: Find (if exist) $k$-pairwise ...
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Cheapest Insertion is $2$-approximation for TSP
Consider the Cheapest Insertion Algorithm on a complete graph with $n$ vertices, where each edge $uv$ has a weight $w(uv)$, and the weights satisfy the triangle inequality $w(xz)\leq w(xy)+w(yz)$ for ...
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