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Clique partitions and clique coverings

1988
*
Discrete Mathematics
*

times as fast as the

doi:10.1016/0012-365x(88)90197-5
fatcat:2htnjyhsgvfddlcrxrnikxq2xm
*clique**covering*number, where c is at least l/64. ... Several new tools are presented for determining the number of*cliques*needed to (edge-)*partition*a graph. For a graph on n vertices, the*clique**partition*number can grow et-? ... He asks about*clique**coverings**and**clique**partitions*of T,,. ...##
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Clique covering and clique partition in generalizations of line graphs

1995
*
Discrete Applied Mathematics
*

In this note we show that minimum sets of maximal

doi:10.1016/0166-218x(94)00076-p
fatcat:ifnamlrwazepdkngnazrreppv4
*cliques**covering*, respectively*partitioning*the edge set of a graph can be computed efficiently for certain superclasses of the class of line graphs. 0166P218X ...*Cliques*are complete subgraphs of a graph. ... A*clique**partition*or maxclique*partition*is an edge-disjoint*clique**covering*, respectively maxclique*covering*. Note that such a maxclique*partition*does not always exist. ...##
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Clique coverings and partitions of line graphs

2008
*
Discrete Mathematics
*

*clique*

*partitions*

*and*

*clique*

*covers*of a line graph, Discrete Math. 83 (1990) 49-62]. ... The

*clique*

*covering*(

*partition*) number cc(G) (cp(G)) of G is the minimum size of a

*clique*

*covering*(

*partition*) of G. ... The authors thank a referee who kindly pointed out that these results are in fact old,

*and*likes the alternative proof for the De Bruijn-Erdős Theorem. ...

##
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On the number of distinct minimal clique partitions and clique covers of a line graph

1990
*
Discrete Mathematics
*

A minimal

doi:10.1016/0012-365x(90)90220-c
fatcat:r2rdf3haznga5p24rz32jfvtpe
*clique**covering*(*partition*) P is a*clique**covering*(*partition*) with IPI = cc(G) (IPI = cp(G)). ... The*clique**covering*number cc(G) is the quantity min{ IPI : P is a*clique**covering*of G}; the*clique**partition*number cp(G) is the quantity min{lPl : P is a*clique**partition*of G}. ...##
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Maximal-clique partitions of interval graphs

1988
*
Journal of the Australian Mathematical Society
*

It is shown that if an interval graph possesses a maximal-

doi:10.1017/s1446788700030147
fatcat:nrnxawvl6vakzdrybvfhroivye
*clique**partition*then its*clique**covering**and**clique**partition*numbers are equal,*and*equal to the maximal-*clique**partition*number. ... Moreover an interval graph has such a*partition*if*and*only if all its maximal*cliques*are edge-disjoint . 1980 Mathematics subject classification (Amer. Math. Soc): 05 C 35. ... If the*clique**covering*C has cardinality \C\*and*|CX'| > \C\ for all*clique**coverings*C' of G, then C is called a minimal*clique**covering**and*the*clique**covering*number of G, cc(G), is defined to equal ...##
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Complexity results on graphs with few cliques

2007
*
Discrete Mathematics & Theoretical Computer Science
*

Several classes of graphs which have few

doi:10.46298/dmtcs.387
fatcat:zd566nxpufhvrfaej6urs5l7nm
*cliques*are discussed,*and*the complexity of some*partitioning**and**covering*problems are determined for the class of all graphs which have fewer*cliques*than a ... Other problems, such as the vertex*clique**cover**and*edge*clique**cover*problems remain NP-complete on these classes. ... Acknowledgements This research was partially supported by NSERC*and*iCORE. ...##
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Clique Partitions of Glued Graphs

2010
*
Journal of Mathematics Research
*

Let P

doi:10.5539/jmr.v2n2p104
fatcat:4pq4ibdbqfcjvpm3ixi2cywouu
*and*P be minimum*clique**partitions*of G*and*G − C, respectively. Then P ∪ {C} is a*clique**partition*of G, so cp(G) ≤ |P ∪ {C}|*and*hence cp(G) − 1 ≤ cp(G − C). ... For any glued graphs of G 1*and*G 2 at a clone H, since the union of a minimum*clique**partition*of G 1 (or G 2 )*and*minimum*clique**partition*of G 2 − H (or G 1 − H) provides a*clique**partition*of G 1 ... The*clique**covering*number cc (G)*and**clique**partition*number cp(G) are the smallest cardinality among all*clique**coverings**and**clique**partitions*of G, respectively. ...##
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Clique coverings of graphs V: maximal-clique partitions

1982
*
Bulletin of the Australian Mathematical Society
*

the maximal-

doi:10.1017/s0004972700005414
fatcat:z4b6ulxa5rhvrgdtapbelizgu4
*clique**partition*number*and*its relationship to other*clique**covering**and**partition*numbers. ... I is |C'| t |C| for all*clique**coverings*C' of G , then C is called a minimum*clique**covering**and*|C| is called the*clique**covering*number of G , denoted by cc(G) . ... The*clique*A ' must be in M because M*covers*H*and*no member of B*covers*A ' . Now M' = (MV4') u {A} is a maximal-*clique**partition*of G*and*hence is either A or B . But A If. ...##
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On the minimum monochromatic or multicolored subgraph partition problems

2007
*
Theoretical Computer Science
*

We investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs, such as

doi:10.1016/j.tcs.2007.04.033
fatcat:6ny45ast65c4bmtbal3nxxbkk4
*cliques*, cycles, trees*and*paths,*partitioning*V (G), depending on the number ... We also present a greedy scheme that yields a (ln m + 1)-approximation for the problem of finding the minimum number of monochromatic*cliques**partitioning*V (G) for a K − 4 -free graph G, where m is the ... Baogang Xu*and*Prof. Shenggui Zhang for helpful discussions*and*the referees for their helpful suggestions. ...##
###
Clique Cover Width and Clique Sum
[article]

2015
*
arXiv
*
pre-print

When G is the

arXiv:1502.06165v2
fatcat:raa66usvvvbirpwxt2p3kyfjaq
*clique*sum of G_1*and*G_2, we prove that CCW(G) < 3/2(CCW(G_1) + CCW(G_2)). ... The*clique**cover*width of G, denoted by CCW(G), is the minimum value of the bandwidth of all*clique**cover*graphs of G. ... ., q, we define the distance of strips S i*and*S j , in the*partition*P (C, B), to be |j − i|. Proposition 2.1 Let C be a*clique**cover*in G*and*let B be a block in C. ...##
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Recent examples in the theory of partition graphs

1993
*
Discrete Mathematics
*

Robertson

doi:10.1016/0012-365x(93)90520-4
fatcat:spgwc2hn4bh35lzpbjfpua4u7e
*and*K.L. ... Two questions which arose in the study of*partition*graphs are answered by recently discovered examples. An enumeration of the*partition*graphs on ten or fewer vertices is provided. ... A*clique**cover*is minimal if no proper subcollection of its*cliques*is also a*clique**cover*. If a*clique**cover*which satisfies Condition I exists, then it can be taken to be minimal. ...##
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Sigma clique covering of graphs
[article]

2015
*
arXiv
*
pre-print

G,

arXiv:1503.02380v1
fatcat:2vewghybarhgnefn2zuld5gnae
*covering*(resp.*partitioning*) all edges of G such that the sum of sizes of the*cliques*is at most k. ... The sigma*clique**cover*number (resp. sigma*clique**partition*number) of graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of*cliques*of ... The minimum size of a*clique**covering*, a biclique*covering*, a*clique**partition**and*a biclique*partition*of G are called*clique**cover*number, biclique*cover*number,*clique**partition*number*and*biclique ...##
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On Clique Coverings of Complete Multipartite Graphs
[article]

2018
*
arXiv
*
pre-print

A

arXiv:1809.01443v1
fatcat:oxpa2njnubcytaxbnbxw34bjmi
*clique**covering*of a graph G is a set of*cliques*of G such that any edge of G is contained in one of these*cliques*,*and*the weight of a*clique**covering*is the sum of the sizes of the*cliques*in it. ... The sigma*clique**cover*number scc(G) of a graph G, is defined as the smallest possible weight of a*clique**covering*of G. Let K_t(d) denote the complete t -*partite*graph with each part of size d. ... Research of Gerbner*and*Methuku was supported by the National Research, Development*and*Innovation Office -NKFIH, grant K 116769. ...##
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On a k-clique-join of a class of partitionable graphs

2005
*
Computer Science Journal of Moldova
*

We call a graph G O-graph if there is an optimal coloring of the set of vertices

doaj:2fa135775a1b424085de547a36172c7f
fatcat:hkazvssbyvdntg7yiecoyhm2la
*and*an optimal (disjoint)*covering*with*cliques*such that any class of colors intersects any*clique*. ... In this paper, it has been established the relation to [p,q,r]-*partite*graphs*and*the fact that the O-graphs admit a k-*clique*-join. American Mathematical Society (2000): 05C17. ... Let G = (V, E) be an O-graph*and*(S 1 , ..., S ω ) a*partition*in ω α-stable sets of G*and*(Q 1 , ..., Q α ) a disjoint*covering*with α ω-*cliques*of begin H : H =G; while (∃e = xy ∈ E with {x, y} ⊂ Q ...##
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Clique Cover on Sparse Networks
[chapter]

2012
*
2012 Proceedings of the Fourteenth Workshop on Algorithm Engineering and Experiments (ALENEX)
*

Empirical studies show that our algorithms are both efficient

doi:10.1137/1.9781611972924.10
dblp:conf/alenex/BlanchetteKV12
fatcat:oo2qzwd7s5c57nst5kgn4upxti
*and*practical on actual simulated*and*biological networks,*and*that the*clique**covers*obtained on real networks yield biological insights. ... We consider the problem of edge*clique**cover*on sparse networks*and*study an application to the identification of overlapping protein complexes for a network of binary protein-protein interactions. ... Therefore, we can*partition*the edges in E(I ∪ P ) into F 1*and*F 2 , where F 1 is to be*covered*by*cliques*in X 1 ,*and*F 2 is*covered*by*cliques*in X 2 . ...
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