# Questions tagged [graph-theory]

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

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### Has multiobjective prize collecting steiner tree or TSP been studied?

Suppose we have a graph $G$ a root $r$ and each node $v$ has some amount of $c_{v,i}$ of each resource $i$. I connect a set of nodes that maximizes the minimum amount of any resource using at most a ...
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### Induced subgraphs with interface

I am interested in graphs with interfaces, i.e. graphs with distinguished vertices that can be plugged into a larger context. I call "island" of a graph $G$ a subgraph $I$ such that there ...
40 views

### On a modular inverse graph construction

Given a balanced bipartite graph $G_1$ on $2n$ vertices on the condition $PM(G_1)\equiv1\bmod2$, an integer $i$ of size $\Omega(n\log n)$, can we find a balanced bipartite graph $G_2$ on $poly(n)$ ...
1 vote
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1 vote
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### Minimum number of triangles required to cover a complete graph?

Let $K_n$ be a complete graph, I am interested in knowing the minimum number of triangles required to get a edge cover of $K_n$. In case there is no closed-form solution to this problem, then I would ...
63 views

### Is every 4-claw-free graph a bounded degree graph?

I am looking of some graph properties of 4-claw free graph, where neighborhood of every vertex has independent set of size at most 3. As per my observations, this type of independent set size ...
169 views

### property of minimal triangulations

A graph is chordal if every cycle on four or more vertices contains a chord i.e. an edge between non-adjacent vertices of the cycle. A triangulation (or chordalization) of a graph $G=(V,E)$ is the ...
299 views

### A non-trivial combinatorial optimization

So I stumble over this problem in which I couldn't find anything similar in the literature. I am not even sure if it is NP-hard or solvable in polynomial time. Any thought or suggestion would be ...
131 views

### Error in TAOCP 4a on the bipartite graph constructed from a hypergraph

The first sentence on page 33 of Donald Knuth's The Art of Computer Programming (TAOCP) Vol. 4a reads: Furthermore, a hypergraph is equivalent to a bipartite graph with vertex set $V \cup E$ and ...
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### What is the fastest algorithm for computing exact network reliability?

In the network reliability problem, we are given an undirected graph $G$ on $n$ vertices and a parameter $p\in (0,1)$, and are tasked with determining the probability that $G$ becomes disconnected (i....
1 vote
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### Upper bound on the number of maximal paths in rooted intransitive DAGs

Let $D(V, A)$ be a DAG. Definition 0: We name a path between two nodes $i$ and $j$ as an $i$-$j$-path. Definition 1: Let $p$ be a path, we call $|p|$ the path length, representing the number of arcs ...
169 views

### 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 ...
164 views

### Given a weighted graph with $pk$ nodes find a min weight forest with $p$ components each containing exactly $k$ nodes

Given a weighted graph with $pk$ nodes find a min weight forest with $p$ components each containing exactly $k$ nodes. Does this have a constant approximation? ($p,k$ and the graph are all part of the ...
126 views

### What is the smallest graph of treewidth $k$ having less edges than the $(k+1)$-clique?

Treewidth is a graph parameter measuring how close a graph is to being a tree. I am interested in what is the minimal number of edges required for a graph to have treewidth $k$. A natural family of ...
2k views

### How bad can the greedy coloring (list color) for the c-chromatic number of graph be?

c-chromatic number is defined in the paper Partitions of graphs into cographs. It asks for the minimum number of colors used to color vertices such that each color class is a cograph. Cograph is a P4-...
47 views

1 vote
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### Stable/Robust Traveling Salesman Approximation Methods

I was wondering if there are TSP approximations that are "stable". More specifically, consider the set $G = x_1, ..., x_n$ and the set $G^* = G \cup x_{n+1}$, where $x_i$ are points in $R^d$....
17 views

### Any value in a formula that calculates (not look up) the 'order' of a 'Independent Edge Set' OR a 'I.E.S.' given an 'order' on complete graphs?

Any value or interest in a formula that calculates (not look up) the 'integer order' of a given 'Independent Edge Set' OR given an 'Independent Set' calculates the 'integer order' on Complete Graphs? ...
488 views

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### Name for a cyclic path in a graph that visits every vertex while minimizing the maximum number of times a given vertex is revisited?

Me and my colleague are interested in whether anyone has looked into a generalization of Hamiltonian cycles where vertices can be revisited, but we want to minimize the maximum number of times a given ...
1 vote
260 views

### 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 ...
1 vote
628 views

### Generate TSP instances with known optimal

Is there a known (polynomial in number of nodes) algorithm to generate TSP instances with known optimal value? The idea is to be able to generating arbitrary large instances with known optimal value,...
76 views

### A variation of the longest path problem

What about finding a path of maximum length in a given graph which may contain cycles, with the constraint that a vertex (or an edge) can be visited at most X (say 2 or 3) times ? EDIT: X would be ...
66 views

### How to reduce a code down to its configuration

I have built a system where from atomic information of a UI code I could generate a framework specific code. Here is the concept https://github.com/imvetri/ui-editor. For example, the user of this ...
1 vote
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### Generalization of the Hamiltonian path problem on Grid Graphs

Fix a cost to each of these actions: move up, move down, move left, move right. I.e. fix some function $f: \{\text{move up, move down, move left, move right}\} \to \mathbb N$. Define the following ...
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### Complexity of greedy coloring

I was looking at some heuristics for coloring and found this book on Google books: Graph Colorings By Marek Kubale They describe the Greedy algorithm as follows: ...
84 views

### 5-color graph and minor

We have a 5-color graph G without 5-clique. The question is: is there a minor H of G that is a 5-clique? Here the minor definition. With "5-color graph G" I mean $\chi (G)=5$.
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### Upper Bound for distance-two chromatic number in terms of maximum degree

Let us consider simple,finite, undirected graphs. A distance-two colouring of a graph $G$ is a function $f:V(G)\to\{1,2,\dots\}$ such that $f(u)\neq f(v)$ whenever $dist_G(u,v)\leq 2$. A distance-two ...
1 vote
33 views

### Application LCL definition to vertex coloration

I'm reading the article "What can be computed locally?" by Naor & Stockmeyer and I struggle to understand the definition of an LCL they gave. Here is an extract: (page 2) An Locally ...
88 views

### Independent set queries with preprocessing

Suppose we have a sparse undirected graph $G = (V, E)$ with $|E| = O(|V|)$, and we want to process it and then answer queries of the following type: given a set $A$, is it an independent set in the ...
1 vote
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### What is known about the complexity of Network Diversion?

In the Network Diversion problem, we are given an undirected graph $G$ on $n$ vertices, with specified nodes $s$ and $t$ and specified edge $e$, and a positive integer $k$, and are tasked with ...
205 views

### Transitive reduction not provably minimal [closed]

Working on finding minimal equivalent graphs, which unlike transitive reductions only allows for edge removals from the original graph. I was under the impression that if you allow for new edges to be ...
76 views

### What's the exact complexity of a DFS if we revisit nodes?

By "revisit nodes," I mean if we didn't maintain a set of nodes we have visited. So the sum I'm examining is just the number of paths from a root to a node, across all roots and nodes. We'll ...
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 ...
Here, by 3D-grid of sidelength $n$ I mean the graph $G=(V,E)$ with $V= \{1,\ldots,n\}^3$ and $E=\{( (a,b,c) ,(x,y,z) ) \mid |a-x|+|b-y|+|c-z|=1 \}$. I known how to get the treewidth of $n*n$ grid is ...