# Tagged Questions

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### Two paper appear to imply collapse via coloring $P_5$-free graphs

Found this from graphclasses.org. Two papers give conflicting results for coloring $P_5$-free graphs which appear to imply $P=NP$. From Polynomial-time algorithm for vertex k-colorability of P_5-...
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### Name the graph class: Disjoint union of a clique and an independent set

Let $G$ be a graph which is the disjoint union of a clique and an independent set, i.e. $$G = K_{n_1} + \overline{K_{n_2}} = K_{n_1} + I_{n_2} .$$ The graph class of all such graphs is characterized ...
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### Weird claim of graphclasses about complexity of domination

EDIT this got 'fixed' on graphclasses, as per answers/comments, so you might not reproduce it, unless you have their earlier database, which is publicly available via sage - http://sagemath.org. ...
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### Is it necessary to call matrix multiplication $n$ times to find a claw

A claw is a $K_{1,3}$. A trivial algorithm will detect a claw in $O(n^4)$ time. It can be done in $O(n^{\omega+1})$, where $\omega$ is the exponent of fast matrix multiplication, as follows: take the ...
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### Natural graph class with five excluded subgraphs?

I'm interested in hereditary graph classes characterized by a small number of excluded subgraphs. There are some well-known graph classes that are characterized by three or four obstructions -- ...
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### Can a natural graph problem be universally hard?

Is there a natural $\mathsf{NP}$-complete graph problem, which remains $\mathsf{NP}$-complete even when it is restricted to any polynomial-time recognizable graph class? To avoid degenerated cases, ...
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### Is there any triangle-free, star-cutset-free, circle graph, with more than n edges?

I'm trying to find a graph with those properties for my studies, but unfortunately I can't find such graph. Does anyone know if there is that graph, or why is it impossible to exist?
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### Edge Cut of interval graphs

On interval graphs, minimal vertex separators are well understood: they are cliques, there are no more than $n$ ones. However, when we turn to the minimal edge cut, my search found no even one single ...
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### Does this graph class have a name ?

It's formulated by extending threshold graphs. Given a threshold graph $(C,I)$ where $C$ is the clique and $I$ is the independent set, my extension is as follows: Each vertex $v\in I$ can be replaced ...
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### Is any chordal graph an incomparability graph?

I was confused by Wikipedia's definitions of "chordal graph", "interval graph", "string graph", "comparability graph", "incomparability graph" and the complements of these. Wikipedia says "The ...
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### Hard problems on subclasses of planar cubic bipartite graphs

Several hard graph problems remain hard on planar cubic bipartite graphs. They include Hamiltonian cycle problem and perfect P3 matching problem. I'm looking for a reference on interesting subclasses ...
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### Do “outer-bounded-genus” graphs have constant treewidth?

Let $k\in\mathbb{N}$ and denote by $G_k$ the set of all graphs that can be embedded on a surface of genus $k$ such that all vertices are situated on the outer face. For instance, $G_0$ is the set of ...
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### Classes of graphs with easy Hamiltonian cycle but NP-hard TSP

The Hamiltonian Cycle Problem (HC) consists in finding a cycle that goes through all vertices in a given undirected graph. The Travelling Salesman Problem (TSP) consists in finding a cycle that goes ...