A {\it krausz $(k,m)$-partition} of a graph $G$ is the partition of $G$ into
cliques, such that any vertex belongs to at most $k$ cliques and any two
cliques have at most $m$ vertices in common. The {\it $m$-krausz} dimension
$kdim_m(G)$ of the graph $G$ is the minimum number $k$ such that $G$ has a
krausz $(k,m)$-partition. 1-krausz dimension is known and studied krausz
dimension of graph $kdim(G)$.
  In this paper we prove, that the problem $"kdim(G)\leq 3"$ is polynomially
solvable for chordal graphs, thus partially solving the problem of P. Hlineny
and J. Kratochvil. We show, that the problem of finding $m$-krausz dimension is
NP-hard for every $m\geq 1$, even if restricted to (1,2)-colorable graphs, but
the problem $"kdim_m(G)\leq k"$ is polynomially solvable for $(\infty,1)$-polar
graphs for every fixed $k,m\geq 1$

Glebova, Olga

Metelsky, Yury

Skums, Pavel

English

arXiv

Секция 10. Теоретическая информатикаA krausz (k, m)-partition of a graph G is the partition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a krausz (k, m)-partition. 1-krausz dimension is known and studied krausz dimension of graph kdim(G). In this paper we prove, that the problem “kdim(G) ≤ 3” is polynomially solvable for chordal graphs, thus partially solving the problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding m-krausz dimension is NP-hard for every m  even if restricted to (1, 2)-colorable graphs, but the problem “kdimm(G) ≤ k” is polynomially solvable for (∞, 1)-polar graphs for every fixed k, m ≥ 1

Glebova, O. V.

Metelsky, Yu. M.

Skums, P. V.

BSU Digital Library

Krausz dimension and its generalizations in special graph classes

Электронная библиотека БГУ

 347
KRAUSZ  DIMENSION  AND  ITS  GENERALIZATIONS 
IN  SPECIAL  GRAPH  CLASSES 
 
O. V. Glebova1, Yu. M. Metelsky2, P. V. Skums3 
 
1, 2 Belarusian State University 
Minsk, Belarus 
E-mail: glebovaov@gmail.com, metelsky@bsu.by 
3 Centers for Disease Control and Prevention  
Atlanta, GA, USA 
E-mail: skumsp@gmail.com 
 
A krausz (k, m)-partition of a graph G is the partition of G into cliques, such 
that any vertex belongs to at most k cliques and any two cliques have at most m verti-
ces in common. The m-krausz dimension kdimm(G) of the graph G is the minimum 
number k such that G has a krausz (k, m)-partition. 1-krausz dimension is known and 
studied krausz dimension of graph kdim(G). In this paper we prove, that the problem 
“kdim(G) ≤ 3” is polynomially solvable for chordal graphs, thus partially solving the 
problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding m-
krausz dimension is NP-hard for every ,1≥m  even if restricted to (1, 2)-colorable 
graphs, but the problem “kdimm(G) ≤ k” is polynomially solvable for (∞, 1)-polar 
graphs for every fixed k, m ≥ 1. 
 
Key words: Krausz dimension, intersection graph, linear k-uniform hypergraph, 
chordal graph, polar graph.  
 
INTRODUCTION 
 
In this paper we consider finite undirected graphs without loops and multiple edges. 
The vertex and the edge sets of a graph (hypergraph) G are denoted by V(G) and E(G) re-
spectively. Given a graph G, let G(X) and G  denote, respectively, the subgraph of G in-
duced by a set X ⊆ V(G) and the complement of G.  
A krausz partition of a graph G is the partition of G into cliques (called clusters of the 
partition), such that every edge of G belongs to exactly one cluster. If every vertex of G be-
longs to at most k clusters then the partition is called krausz k-partition. The krausz dimen-
sion kdim(G) of the graph G is the minimum k such that G has krausz k-partition. 
Krausz k-partitions are closely connected with the representation of a graph as the in-
tersection graph of a hypergraph. The intersection graph L(H) of a hypergraph 
H = (V(H), E(H)) is defined as follows:  
1. the vertices of L(H) are in a bijective correspondence with the edges of H; 
2. two vertices are adjacent in L(H) if and only if the corresponding edges have a 
nonempty intersection. 
Hypergraph H is called linear, if any two of its edges have at most one common ver-
tex; k-uniform, if every edge contains k vertices; Helly hypergraph, if for every family of 
 348
hyperedges 1,..., ( )rЕ Е Е H∈  such that i iЕ Е ≠ ∅  for every i, j = 1,…, r we have 
1
.
r
i
i
Е
=
≠ ∅  
The multiplicity of the pair of vertices u, v of the hypergraph H is the number 
{ }( , ) ( ) : , ;m u v E E H u v E= ∈ ∈  the multiplicity m(H) of the hypergraph H is the maximum 
multiplicity of the pairs of its vertices. So, linear hypergraphs are the hypergraphs with the 
multiplicity 1. 
Denote by H∗ the dual hypergraph of H and by H[2] the 2-section of H (i. e. the simple 
graph obtained by transformation each edge into a clique). It follows immediately from the 
definition that  
 L(H) = (H∗)[2] (1) 
(first this relation was implicitly formulated by C. Berge in [2]). This relation implies that a 
graph G has krausz k-partition if and only if it is intersection graph of linear k-uniform hy-
pergraph.  
A graph is called (p, q)-colorable, if its vertex set could be partitioned into p cliques 
and q stable sets. In this terms (1, 1)-colorable graphs are well-known split graphs. 
Another generalization of split graphs is the class of polar graphs. A graph G is called 
polar if there exists a partition of its vertex set  
 ( ) ,V G A B=    A B = ∅  (2) 
(bipartition (A, B)) such that all connected components of the graphs ( )G A  and G(B) are 
complete graphs. If, in addition, α and β are fixed integers, and the orders of connected 
components of the graphs ( )G A  and G(B) are at most α and β respectively, then the polar 
graph G with bipartition (2) is called (α, β)-polar. Given a polar graph G with bipartition 
(2), if the order of connected components of the graph ( )G A  (the graph G(B)) is not re-
stricted above, then the parameter α (respectively β) is supposed to be equal ∞. Thus an ar-
bitrary polar graph is (∞, ∞)-polar, and a split graph is (1, 1)-polar. 
Denote by KDIM(k) the problem of determining whether kdim(G) ≤ k and by KDIM 
the problem of finding the krausz dimension. 
The class of line graphs (intersection graphs of linear 2-uniform hypergraphs, i. e. 
graphs with krausz dimension at most 2) have been studied for a long time. It is character-
ized by a finite list of forbidden induced subgraphs [1], efficient algorithms for recognizing 
it (i. e. solving the problem KDIM(2)) and constructing the corresponding krausz 2-partition 
are also known (see for example [4], [7], [13], [14]). 
The situation changes radically if one takes k = 3 instead of k = 2: the problem 
KDIM(k) is NP-complete for every fixed k ≥ 3 [5]. The case k = 3 was studied in the differ-
ent papers (see [6], [10], [11], [12], [15]), and several graph classes, where the problem 
KDIM(3) is polynomially solvable or remains NP-complete, were found.  
In [5] P. Hlineny and J. Kratochvil studied the computational complexity of the 
krausz dimension in detail. In particular, they proved, that for chordal graphs the problem 
KDIM(k) is NP-complete for every k ≥ 6. 
So, P. Hlineny and J. Kratochvil posed the problem of deciding the complexity of 
KDIM(k) restricted to chordal graphs for k = 3, 4, 5. In this paper we give a partial answer 
to this problem (namely, in the case k = 3).  
 349
Further we consider the natural generalization of the krausz dimension. The krausz 
(k, m)-partition of a graph G is the partition of G into cliques (called clusters of the parti-
tion), such that any vertex belongs to at most k clusters of the partition, and any two clusters 
have at most m vertices in common. As above, the relation (1) implies, that graphs with 
krausz (k, m)-partitions are exactly the intersection graphs of k-uniform hypergraphs with 
the multiplicity at most m. The m-krausz dimension kdimm(G) of the graph G is the mini-
mum k such that G has a krausz (k, m)-partition. The krausz dimension in these terms is the 
1-krausz dimension. In this paper we present some computational complexity results con-
cerning the m-krausz dimension of graph.  
 
FORMULATION  OF  THE  RESULTS 
 
Denote by lc(H) and Δ(H) the length of a longest induced cycle and the maximum 
vertex degree of a graph H, respectively.  
Lemma 1. There exists a polynomial-time algorithm, which takes a chordal graph G 
as an input and constructs the graph H with ∆(H) ≤ 18 and lc(H) ≤ 6 such that kdim(G) ≤ 3 
if and only if kdim(H) ≤ 3. 
P. Hlineny and J. Kratochvil proved in [5], that the problem KDIM is polynomially 
solvable for graphs with bounded treewidth. The following relation was proved in [3]: if 
lc(H) ≤ s + 2 and ∆(H) ≤ ∆, then treewidth(H) ≤ ∆ (∆ − 1)s − 1. These two facts together with 
Lemma 1 imply the following statement.  
Theorem 2. The problem KDIM(3) is polynomially solvable for chordal graph. 
Denote by KDIMm the problem of determining the m-krausz dimension of graph, by 
KDIMm (k) the problem of determining whether kdimm(G) ≤ k and by mkL  the class of graphs 
with a krausz (k, m)-partition. It was proved in [8] that the class 3
mL  could not be character-
ized by a finite set of forbidden induced subgraphs for every m ≥ 2, but the complexity of 
the problem KDIMm for an arbitrary m was not established yet. We proved the following: 
Theorem 3. The problem KDIMm is NP-hard for every m ≥ 1, even if restricted to the 
class of (1, 2)-colorable graphs. 
The class mkL  is hereditary (i. e. closed with respect to deleting the vertices) and there-
fore can be characterized by the infinite list of forbidden induced subgraphs. We proved the 
following: 
Theorem 4. There exists a finite set F of forbidden induced subgraphs such that an 
(∞, 1)-polar graph G belongs to the class mkL  if and only if no induced subgraph of G is 
isomorphic to an element of F. 
Corollary 5. The problem KDIMm(k) is polynomially solvable in the class of (∞, 1)-
polar graphs for every fixed k, m ≥ 1. 
In particular, Corollary 5 generalizes the result of [5] and [9], that for every fixed k 
the problem KDIM(k) is polynomially solvable for split graphs.  
 
LITERATURE 
 
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Krausz dimension and its generalizations in special graph classes

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