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MAX for $k$-independence in multigraphs
- Publication Year :
- 2018
-
Abstract
- For a fixed positive integer $k$, a set $S$ of vertices of a graph or multigraph is called a $k$-independent set if the subgraph induced by $S$ has maximum degree less than $k$. The well-known algorithm MAX finds a maximal $k$-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than $k$. We give an efficient procedure that determines, for a given degree sequence $D$, the smallest cardinality $b(D)$ of a $k$-independent set that can result from any application of MAX to any loopless multigraph with degree sequence $D$. This analysis of the worst case is sharp for each degree sequence $D$ in that there exists a multigraph $G$ with degree sequence $D$ such that some application of MAX to $G$ will result in a $k$-independent set of cardinality exactly $b(D)$.<br />Comment: 16 pages, 5 figures
- Subjects :
- Mathematics - Combinatorics
05C69 (Primary) 05B40 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1807.04997
- Document Type :
- Working Paper