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# Questions tagged [graph-theory]

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

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0answers
29 views

### end node with unique path length from the start node in a DAG

Let $G(V,E)$ be a directed acyclic graph with all edge weights set to one and $s\in V$ be the start node, $E \in V\backslash s$ be the set of end nodes. My problem is to find an end node $e\in E$ ...
0answers
34 views

### Does every graph of clique-width 3 have a large induced subgraph of clique-width 2?

Is there a constant $\alpha>0$ such that every graph $G$ of clique-width $3$ and order $n$ has an induced subgraph of order at least $\alpha n$ and clique-width at most $2$ (in other words, the ...
0answers
39 views

### 3-hitting set iterative compression

I have a question which i tried to solve without success. I need to prove that if 3-Hitting Set can be solved in time $2^kn^{O(1)}$,then 4-Hitting Set can be solved in time $3^kn^{O(1)}$. There is a ...
0answers
39 views

### Claw-free graph linear kernel [closed]

I'm having a hard time solving the problem below: In Claw-free problem, we are given a graph G and $k$, and the objective is to decide whether there exists a subset S $\subseteq$ V (G) of size at most ...
1answer
139 views

### Is the difference between the acyclic chromatic number and the star chromatic number unbounded?

Is $\chi_s(G)-\chi_a(G)$ unbounded in general graphs? I thought $\chi_s(G)-\chi_a(G)$, the difference between the star chromatic number and the acyclic chromatic number, is unbounded for general ...
0answers
25 views

### Randomized algorithm for finding Minimum feedback vertex set

Algorithm FVS(G, k): If k < 0, return ”NOT FOUND” If G is acyclic (i.e., a forest), return  While there exists a vertex 𝑢 of degree at most 2: If deg(u) = 1, remove u If deg(u) = 2, i.e. u's ...
1answer
212 views

### Find research partner (profession and beginner)

I've 10 years of industrial work, but in my free time, I do research, write papers to conferences, help to teach to my old friend at the university and I even did a Ph.D. full-time program. Now, I've ...
0answers
88 views

### Has this notion of connectivity in edge-colored graphs been studied?

Consider a simple graph $G$ where each edge is either red or blue. I'm interested in the following notion of connectivity: Two vertices $u$ and $v$ are said to be connected if there is a path ...
1answer
122 views

### Detect if a graph has a $k$ cycle in space complexity $O((\log k)^d)$ for fixed $d \geq1$

For a graph $G$, I want to test if it contains a cycle of length $k$, for some $k$ much smaller than $|G|$. I am interested in particular in an algorithm with low space complexity. The cycle need not ...
1answer
51 views

### Do such instances always admit a 3D matching?

I want to know whether the following kinds of special instances of the 3D Matching problem are yes" instances, i.e., admit a 3D matching. We are given 3 sets $A,B,C$ containing $m$ elements ...
1answer
145 views

### TSP with “enemy” nodes

I am curious if the following variation of the traveling salesman problem (TSP) (or a vehicle routing problem (VRP) version) occurs in the literature and has a name I could search for. The story/idea ...
1answer
118 views

### Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist. Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
1answer
125 views

### What are the known classes of undirected graphs such that every graph belonging to that class is guaranteed to have a Hamiltonian Path?

One trivial class of graphs is the class consisting of complete graphs or complete bipartite graphs with equal sized partitions. I would love to know if more such classes exist.
1answer
89 views

### upper bound on the total number of fixed-length paths in an acyclic graph [closed]

I was wondering if there is an upper bound on the total number of fixed-length paths (path length from 1 to $n-1$ given $n$ nodes) in an acyclic graph (not directed) of $n$ nodes? If so, can you point ...
1answer
163 views

### Does such a bipartite graph exist?

In the course of my studies on graphs I sometimes use gadgets. I recently came upon a need for a certain bipartite graph with the following properties, and I am wondering if anyone knows if such a ...
0answers
85 views

### Complexity of counting 3-colourings of planar bounded degree graphs

The following are known: It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1]. For all $d\geq 3$, it is #P-complete to count the number ...
0answers
148 views

### How to find the second smallest cut in a graph?

For an undirected graph, how do we find the second smallest $s,t-$cut(s) for some $s,t\in V$? What's the time complexity of this computation? What if we only cared about finding a cut of size $p+1$, ...
0answers
32 views

### Listing Eulerian orienations with special properties

An orientation of a simple undirected graph is said to be Eulerian if every vertex has the same number of in-coming edges and out-going edges (i.e., in-degree($v$)=out-degree($v$) for all $v\in V(G)$)....
1answer
201 views

### Complexity of relaxed edge colouring

A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
0answers
108 views

### Decomposing graph homomorphisms

A homomorphism $h: G\to H$ from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to those of $H$ which preserves edges, that is, if $(x,y)$ is an edge of $G$ then $(h(x),h(y))$ is an ...
0answers
45 views

2answers
124 views

### Maximum cliques of the transitive closure of a chordal DAG

Let $G=(V,A)$ be a directed acyclic graph, for which the underlying undirected graph is chordal (so that every induced cycle in the underlying undirected graph is a triangle). It is known that in a ...
0answers
144 views

### Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
1answer
40 views

### Bipartite graph projections, with threshold

Let $G=(\top,\bot,E)$ be a bipartite graph: $E\subseteq \top\times\bot$. The projections $G_\bot = (\bot,E_\bot)$ and $G_\top = (\top,E_\top)$ of $G$ are defined as follows: two vertices are linked ...
2answers
84 views

### Graph labelling where vertices with a common neighbour get different labels

Is the following notion of graph labeling (let me call it special labeling) ever studied in the literature? A special labelling of a (simple undirected) graph $G$ is a function $f:V(G)\to\mathbb{N}$ ...
0answers
27 views

### Unique naming/labeling of $40$-node strongly regular graphs

Brendan McKay's webpage lists all possible $40$-node strongly regular graphs. Is there a standard way to name them uniquely?
1answer
34 views

### Maximum weight matching with classes of edges in a multi-edge bipartite graph

Posted a similar question in mathoverflow, have tried to reduce this to Ford Fulkerson, but been stuck. Thought I'd ask TCS community to see if there are any ideas from individuals, here. Consider a ...
2answers
341 views

0answers
163 views

### Is this node permutation optimization NP-Hard?

Let $G=(V,E)$ be an undirected graph and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{succ}_{\pi}(v)$ the set of neighbors of $v$ that occur after $v$ in ...
1answer
127 views

### Random Cerny Conjecture

For simplicity, all DFAs will be using the binary alphabet $\{0,1\}$. Let $M$ be a synchronizable DFA. We let $p(M,n)$ be the probability that a random $x\in \{0,1\}^n$ will synchronize $M$. We define ...
2answers
95 views

1answer
110 views

### When does a bipartite graph have bounded treewidth?

As the title says, I want to know when the treewidth of a bipartite graph is bounded by a constant. What families of graphs are both bipartite and bounded treewidth? More generally, I would like to ...
0answers
36 views

### Are there classes where all Eulerian orientations can be listed in polynomial time?

Is there is a subclass of regular graphs (say 4-regular graphs) for which there is a polynomial time algorithm to list all Eulerian orienations? An Eulerian orientaiton of an (undirected simple) graph ...