Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1207.0255

Extending Partial Representations of Subclasses of Chordal Graphs

Pavel Klavik·Jan Kratochvil·Yota Otachi·Toshiki Saitoh
View original source ↗Read source document ↗Export metadata ↓

Abstract

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T' and some pre-drawn subtrees of T'. It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e, keeps the pre-drawn subtrees unchanged). We consider four modifications of T' and get vastly different problems. In some cases, it is interesting to consider the complexity even if just T' is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization. We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components and the size of the tree T'. We describe an interesting relation with integer partition problems. The problem Partition is used for all NP-completeness reductions. The extension of interval graphs when the space in T' is limited is "equivalent" to the BinPacking problem.

Explore related subjects

cs.CC

Keep this discovery

BibTeXRIS

Pavel Klavik, Jan Kratochvil, Yota Otachi, Toshiki Saitoh. 2013-05-20. Extending Partial Representations of Subclasses of Chordal Graphs. https://arxiv.org/abs/1207.0255

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗