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Sensor Location Problem for a Multigraph

Pilipchuk, L. A., Vishnevetskaya, T. S., Pesheva, Y. H. (2013)

Mathematica Balkanica New Series

MSC 2010: 05C50, 15A03, 15A06, 65K05, 90C08, 90C35We introduce sparse linear underdetermined systems with embedded network structure. Their structure is inherited from the non-homogeneous network ow programming problems with nodes of variable intensities. One of the new applications of the researched underdetermined systems is the sensor location problem (SLP) for a multigraph. That is the location of the minimum number of sensors in the nodes of the multigraph, in order to determine the arcs ow...

Sequences of Maximal Degree Vertices in Graphs

Khadzhiivanov, Nickolay, Nenov, Nedyalko (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 05C35.Let Γ(M ) where M ⊂ V (G) be the set of all vertices of the graph G adjacent to any vertex of M. If v1, . . . , vr is a vertex sequence in G such that Γ(v1, . . . , vr ) = ∅ and vi is a maximal degree vertex in Γ(v1, . . . , vi−1), we prove that e(G) ≤ e(K(p1, . . . , pr)) where K(p1, . . . , pr ) is the complete r-partite graph with pi = |Γ(v1, . . . , vi−1) Γ(vi )|.

Séries de croissance et polynômes d'Ehrhart associés aux réseaux de racines

Roland Bacher, Pierre de La Harpe, Boris Venkov (1999)

Annales de l'institut Fourier

Étant donnés un système de racines R d’une des familles A, B, C, D, F, G et le groupe abélien libre qu’il engendre, on calcule explicitement la série de croissance de ce groupe relativement à R . Les résultats s’interprètent en termes du polynôme d’Ehrhart de l’enveloppe convexe de R .

Set colorings in perfect graphs

Ralucca Gera, Futaba Okamoto, Craig Rasmussen, Ping Zhang (2011)

Mathematica Bohemica

For a nontrivial connected graph G , let c : V ( G ) be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v V ( G ) , the neighborhood color set NC ( v ) is the set of colors of the neighbors of v . The coloring c is called a set coloring if NC ( u ) NC ( v ) for every pair u , v of adjacent vertices of G . The minimum number of colors required of such a coloring is called the set chromatic number χ s ( G ) . We show that the decision variant of determining χ s ( G ) is NP-complete in the general case, and show that χ s ( G ) can be...

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