Computer Science / Discrete Mathematics Quantitative Biology / q-bio.QM

On restrictions of balanced 2-interval graphs

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#Discrete Mathematics #Quantitative Biology #Computer Science

📝 Original Info

  • Title: On restrictions of balanced 2-interval graphs
  • ArXiv ID: 0704.1571
  • Date: 2008-02-04
  • Authors: Researchers from original ArXiv paper

📝 Abstract

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, ...

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Deep Dive into On restrictions of balanced 2-interval graphs.

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, …

📄 Full Content

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Ö Ð Ø ÒØ ÖÚ Ð× ÒØ Ö× Ø¸×Ó l(v 2,l ) < l(v 1,l ) < l(v ′ 2,l ) < l(v ′ 1,l )º

• Ø Ö l(v σ(2),l ) < l(v σ(1),l ) < l(v ′ σ(2),l ) < l(v ′ σ(1),l ) Ò l(v ′ σ(2),r ) < l(v ′ σ(1),r ) < l(v σ(2),r ) < l(v σ(1),r )•

• Ø Ö l(v σ(x-1),l ) < . . . < l(v σ(1),l ) < l(v ′ σ(x-1),l ) < . . . < l(v ′ σ(1),l ) Ò l(v ′ σ(x-1),r ) < . . . < l(v ′ σ(1),r ) < l(v σ(x-1),r ) < . . . < l(v σ(1),r )¸½ ¾

• ÓÖ Ø ×ÝÑÑ ØÖ × l(v σ(1),l ) < . . . < l(v σ(x-1),l ) < l(v ′ σ(1),l ) < . . . < l(v ′ σ(x-1),l ) Ò l(v ′ σ(1),r ) < . . . < l(v ′ σ(x-1),r ) < l(v σ(1),r ) < . . . < l(v σ(x-1),r )º

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Ì Ò Û Ö ØÐÝ Ø Ò ÕÙ Ð Ø ×

• l(v 1 r ) < l(v σ ′ (x),l ) < . . . < l(v σ ′ (j+1),l ) ≤ l(v x,l ) < l(v σ ′ (j-1),l ) < . . . < l(v σ ′ (1),l ) < l(v ′ σ ′ (x),l ) < . . . < l(v ′ σ ′ (j+1),l ) < l(v ′ σ ′ (j-1),l ) < . . . < l(v ′ σ ′ (1),l )

• l(v ′ σ ′ (x),r ) < . . . < l(v ′ σ ′ (j+1),r ) < l(v ′ σ ′ (j-1),r ) < . . . < l(v ′ σ ′ (1),r ) < l(v σ ′ (x),r ) < . . . < l(v σ ′ (j+1),r ) < l(v σ ′ (j-1),r ) < . . . < l(v σ ′ (1),r )

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