A survey of graph layout problems
ACM Computing Surveys (CSUR)
The graph sandwich problem for 1-join composition is NP-complete
Discrete Applied Mathematics
Embeddings of k-Connected Graphs of Pathwidth k
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
On the Complexity of (k, l)-Graph Sandwich Problems
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
VLSI layout of trees into grids of minimum width
Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures
On decision and optimization (k, l)-graph sandwich problems
Discrete Applied Mathematics
Embeddings of k-connected graphs of pathwidth k
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
The sandwich problem for cutsets: clique cutset, k-star cutset
Discrete Applied Mathematics - Special issue: Traces of the Latin American conference on combinatorics, graphs and applications: a selection of papers from LACGA 2004, Santiago, Chile
Discrete Applied Mathematics
On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs
Theoretical Computer Science
Some approximation algorithms for the clique partition problem in weighted interval graphs
Theoretical Computer Science
Minimal proper interval completions
Information Processing Letters
Random Generation and Enumeration of Proper Interval Graphs
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Embeddings of k-connected graphs of pathwidth k
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Discrete Applied Mathematics
A clustering algorithm for interval graph test on noisy data
WEA'03 Proceedings of the 2nd international conference on Experimental and efficient algorithms
The external constraint 4 nonempty part sandwich problem
Discrete Applied Mathematics
Theoretical Computer Science
Minimal proper interval completions
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Minimum clique partition problem with constrained weight for interval graphs
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Approximability of the path-distance-width for AT-free graphs
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Integral mixed unit interval graphs
Discrete Applied Mathematics
Approximating the path-distance-width for AT-free graphs and graphs in related classes
Discrete Applied Mathematics
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We study two related problems motivated by molecular biology. Given a graph $G$ and a constant $k$, does there exist a supergraph $G'$ of $G$ that is a unit interval graph and has clique size at most $k$? Given a graph $G$ and a proper $k$-coloring $c$ of $G$, does there exist a supergraph $G'$ of $G$ that is properly colored by $c$ and is a unit interval graph? We show that those problems are polynomial for fixed $k$. On the other hand, we prove that the first problem is equivalent to deciding if the bandwidth of $G$ is at most $k-1$. Hence, it is NP-hard and $W[t]$-hard for all $t$. We also show that the second problem is $W[1]$-hard. This implies that for fixed $k$, both of the problems are unlikely to have an $O(n^\alpha)$ algorithm, where $\alpha$ is a constant independent of $k$. A central tool in our study is a new graph-theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph.