Kartynnik, Yury

Ryzhikov, Andrew

BSU Digital Library

GRAPHS WITH EQUAL DISTANCE PARAMETERS
Yury Kartynnik1, Andrew Ryzhikov2
1Belarusian State University, Faculty of Applied Mathematics and Informatics
4 Nezavisimosti Ave, 220030, Minsk, Belarus
kartynnik@bsu.by
2 United Institute of Informatics Problems, National Academy of Sciences, Belarus
6 Surganov str., 220072, Minsk, Belarus
ryzhikov.andrew@gmail.com
1. Introduction. The concepts of distance packing and covering in graphs was introduced by
Meir and Moon in [1]. We consider finite, simple, undirected graphs without loops and multiple
edges. A set P of vertices in a graph is called a k-packing (or a k-independent set) if the distance
between any two distinct vertices in this set is larger than k. The maximum size of the k-packings
in a graph G is called the k-packing number of G and is denoted by k(G). A set D of vertices in a
graph G is called a k-covering (or a k-domination set) if for any vertex v in V (G) there is a vertex
in D at a distance no more than k from v. The minimum size of the k-domination sets in a graph
G is called the k-domination number of G and is denoted by k(G). A set I of vertices in a graph
is called a k-independent domination set if it is both a k-packing and a k-covering. The minimum
size of the k-independent domination sets in a graph G is called the k-independent domination
number of G and is denoted by ik(G). For every graph G, the inequality k(G)  ik(G)  k(G)
holds.
The relation between the distance packing, domination and independent domination numbers
has been widely studied in the literature. In [1] it is shown that the equality k(T ) = 2k(T ) holds
for any tree T . In [2] this equality is proved for a larger class of sun-free chordal graphs, which
includes line graphs of trees, interval graphs and powers of block graphs. In [3] the graphs with
equal k-packing and 2k-packing numbers are characterized. This characterization implies a simple
polynomial recognition algorithm for such graphs.
2. Recognition of k-equipackable graphs. A graph G is called k-equipackable if ik(G) =
k(G). For k = 1 such graphs have been widely studied under the name well-covered, see the
survey by Plummer [4]. In [5] it is shown that deciding whether a graph is not k-equipackable
is an NP-complete problem. Lesk and Plummer [6] obtained that the recognition of line 1-
equipackable graphs is a polynomially solvable problem. In [7] it is proved that recognizing
non-2-equipackable line graphs is an NP-complete problem. Our following result establishes the
computational complexity for the problem of recognizing k-equipackable line graphs for k  2.
Theorem 1. Deciding whether a given line graph is not k-equipackable is an NP-complete
problem for any fixed k  2.
Corollary 1. Let G be a line graph. Deciding whether Gk is not well-covered is an NP-complete
problem for any fixed k  2.
3. Subclasses of k-equipackable graphs. Let k be a positive integer and Rk be the class of
graphs with k(G) = 2k(G) for every G 2 Rk. In [3] a simple characterization of the class Rk is
obtained. In [2] it is proved that for every sun-free chordal graph G the equality k(G) = 2k(G)
holds. Using this results, we obtain the following characterization.
Theorem 2. The following statements are equivalent for a sun-free chordal graph G:
1) k(G) = k(G);
2) G 2 Rk.
Corollary 2. The problem of recognizing sun-free chordal graphs G having k(G) = k(G) is
polynomially solvable.
Using the characterization of the graphs with equal k-packing and 2k-packing numbers from
[3], we obtain the following.
Theorem 3. Every graph in Rk is k-equipackable.
Thus, all the sun-free chordal graphs with k(G) = k(G) are k-equipackable. It is an open
question whether there are any other k-equipackable sun-free chordal graphs.
We thank Yury L. Orlovich for stating the question of recognizing the complexity of k-
equipackable line graphs and multiple useful comments and suggestions during the course of this
work.
This work has been partially supported by the Belarusian BRFFR grant (Project F15MLD-
022).
References
1. Meir A., Moon J.W. Relation between packing and covering of a tree // Pacific Journal of Mathematics.
1975. V. 61. P. 225–233.
2. Chang G.J., Nemhauser G.L. The k-domination and k-stability problems on sun-free chordal graphs
// SIAM Journal on Algebraic Discrete Methods. 1984. V. 5. No. 3. P. 332–345.
3. Joos F., Rautenbach D. Equality of distance packing numbers // Preprint, http://arxiv.org/abs/1402.6129.
4. Plummer M.D. Well-covered graphs: A survey // Quaestiones Mathematicae. V. 16. No. 3. P. 253-
287.
5. Favaron O., Haynes T.W., Slater P.J. Distance-k independent domination sequences // The Journal
of Combinatorial Mathematics and Combinatorial Computing. 2000. V. 33. P. 225-237.
6. Lesk M., Plummer M.D., PulleyblankW.R. Equi-matchable graphs. In: Graph theory and combinatorics,
Academic Press, London, 1984. P. 239–254.
7. Baptiste P., Kovalyov M. Y., Orlovich Y. L., Werner F., Zverovich I. E. Graphs with maximal induced
matchings of the same size // Proc. of 14th IFAC Symposium on Control problems in Manufacturing. 2012.
P. 518–523.


Graphs with equal distance parameters

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Graphs with equal distance parameters

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