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### Extremal connected graphs for independent domination number

Robert C. Brigham, Julie R. Carrington, Richard P. Vitray
2004 Discrete Mathematics
For certain situations, one of which occurs when n is a perfect square, the extremal graphs have a particularly simple structure.  ...  A general characterization of connected graphs on n vertices having the maximum possible independent domination number of n + 2 − 2 √ n is given.  ...  Most extremal graphs do not satisfy conditions (a) -(d) of Lemma 19. We immediately obtain a structural characterization of these graphs.  ...

### Overlapping of functions

F. V. Guseinov
1997 Mathematical Notes
The definition of an extreme ray of a set A is similar.  ...  L -L and for any extreme point -2 of the set epi f** fl L -L the graph of the function contains an enveloping subset of -~ + L.  ...

### Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

Yufei Zhao
2017 The American mathematical monthly
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms.  ...  More precisely, in the family of of d-regular graphs, which graph G maximizes/minimizes the quantity i(G)^1/v(G), the number of independent sets in G normalized exponentially by the size of G?  ...  Independent Sets Let G = (V , E) be a graph. An independent set is a subset of the vertices with no two adjacent. Question.  ...

### Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

M SIMONOVITS
1974 Discrete Mathematics
The main result of this paper is that for a special, but rather wide class of "sample graphs", the extremal graphs, i.e, the graphs of n vertices without subgraphs isomorphic to the sample graph and having  ...  This result remains valid even in the case when the' condition "the graph does not contain the sample graph" is replaced by the condition "the graph does not contain the sample graph and its chromatic  ...  We do not distinguish between a chromatic condition and the family of graphs, satisfying this condition. (1) Let A be the family of at least r-chromatic graphs. Then A is a chromatic condition.  ...

### Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

M. Simonovits
1974 Discrete Mathematics
The main result of this paper is that for a special, but rather wide class of "sample graphs", the extremal graphs, i.e, the graphs of n vertices without subgraphs isomorphic to the sample graph and having  ...  This result remains valid even in the case when the' condition "the graph does not contain the sample graph" is replaced by the condition "the graph does not contain the sample graph and its chromatic  ...  We do not distinguish between a chromatic condition and the family of graphs, satisfying this condition. (1) Let A be the family of at least r-chromatic graphs. Then A is a chromatic condition.  ...

### (c-)AND: A new graph model [article]

Mauricio Soto, Christopher Thraves
2013 arXiv   pre-print
In this document, we study the scope of the following graph model: each vertex is assigned to a box in a metric space and to a representative element that belongs to that box.  ...  We give both, a combinatorial and an intersection characterization of the model.  ...  Acknowledgements The authors would like to thank Antonio Fernández Anta and Marcos Kiwi because they are strongly involved in the origins of this study, even more, they contributed with enlightening talks  ...

### Page 1527 of Automation and Remote Control Vol. 55, Issue 10 [page]

1994 Automation and Remote Control
of extreme working regimes for a particular manufacturing complex in the form of a directed graph G(Q, W), where the set Q of vertices of the graph corresponds to initial situations, objective situations  ...  to the operations ut) € T(“+)) and fu) € T() are taken to coincide with the sets p (ut) and p() of input conditions of these operations and have the following forms: Piet!)  ...

### The decycling number of outerplanar graphs

Huilan Chang, Hung-Lin Fu, Min-Yun Lien
2012 Journal of combinatorial optimization
In this paper, we provide a necessary and sufficient condition for an outerplanar graph being upper-extremal.  ...  On the other hand, we find a class S of outerplanar graphs none of which is lower-extremal and show that if G has no subdivision of S for all S ∈ S, then G is lower-extremal.  ...  On the other hand, we provide a sufficient condition for an outerplanar graph being lower-extremal.  ...

### Maximizing the number of maximal independent sets of a fixed size [article]

Chunwei Song, Bowen Yao
2020 arXiv   pre-print
Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs.  ...  In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.  ...  The above extremal value is achieved if and only if the following conditions are simultaneously met. i). G − v is a t-cliques extremal graph of order n − 1.  ...

### On Ramsey—Turán type theorems for hypergraphs

P. Erdős, Vera T. Sós
1982 Combinatorica
condition which decreases the size of the maximal independent set in G,, then the number of edges of the corresponding extremal graphs gets drastically reduced.  ...  E.g. for K; (and for a more general class of graphs) the condition that the largest independent set has size o(n) does not change the situation. We prove Remark.  ...

### On some interconnections between combinatorial optimization and extremal graph theory

Dragos Cvetkovic, Pierre Hansen, Vera Kovacevic-Vujcic
2004 Yugoslav Journal of Operations Research
The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set.  ...  While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various  ...  An instance is a set of graphs specified by the number of vertices and by other graph invariants, i.e. the set of instances is a set of sets of graphs.  ...

### A Generalization of Nemhauser and Trotter's Local Optimization Theorem [article]

Michael R. Fellows, Jiong Guo, Hannes Moser, Rolf Niedermeier
2009 arXiv   pre-print
We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves.  ...  Our generalization of the Nemhauser-Trotter theorem implies that Bounded-Degree Deletion has a problem kernel with a linear number of vertices for every constant d.  ...  A packing P of a graph G is a set of pairwise vertex-disjoint subgraphs of G. A graph has maximum degree d when every vertex in the graph has degree at most d.  ...

### Page 1157 of Automation and Remote Control Vol. 54, Issue 7 [page]

1993 Automation and Remote Control
Thus, applying the dominant rule of choice to graphs of a non-strict partial ordering for all X C A coincides with extrem- ization choice according to a certain set of criteria (Pareto rule), while choice  ...  for graphs of non-strict weak ordering coincides with the conventional extremization choice according to a certain criterion.  ...

### On extreme infinite doubly stochastic matrices

Ryszard Grząślewicz
1987 Illinois Journal of Mathematics
The set of all extreme points of (, g) coincides with the set of all exposed points of ( , ). Proof that F.ia by Obviously each exposed point is extreme. Now let a > 0 be such 1.  ...  We say that the connected component H of the graph G(P) is an extreme tree if H is a tree satisfying the following conditions: (,) H has no e-bitree. (, ,) H has at most one node corresponding to a row  ...

### Page 1014 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 7, Issue 6 [page]

1956 American Mathematical Society. Proceedings of the American Mathematical Society
More precisely: let P be the set of vertices of a connected linear graph y and S a given set of objects of the same cardinality as P.  ...  Consider a connected linear graph y and a function F,(x1, x2, - - - ) where x; is a variable associated to the point P; of the graph. Let 42%2 +--+ 2a;2--- be a given set of numbers.  ...
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