Average distance and generalised packing in graphs

Dankelmann, Peter

doi:10.1016/j.disc.2010.05.006

Cited by 9 publications

(7 citation statements)

References 14 publications

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“…The k-packing number α k (G) of G is the cardinality of a maximum k-packing set [11]. For additional results on k-packing, see [4,10]. Moreover, k-packings are the key ingredients for the concept of the S -packing chromatic number (see [1,3,9] and references therein).…”

Section: Lower Bounds On Gp(g)mentioning

confidence: 99%

A General Position Problem in Graph Theory

Manuel¹,

Klavžar

2018

Bull. Aust. Math. Soc.

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The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

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Section: Lower Bounds On Gp(g)mentioning

confidence: 99%

A General Position Problem in Graph Theory

Manuel¹,

Klavžar

2018

Bull. Aust. Math. Soc.

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“…See [4], [26] for some references. The domination number and the k-distance domination number have moreover been related to the average distance of graphs in the recent papers [7], [8] and [9].…”

Section: Theorem 2 ([5]mentioning

confidence: 99%

Spanning Trees with Many Leaves and Average Distance

DeLaViña¹,

Waller²

2008

Electron. J. Combin.

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In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

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“…Extremal problems for the mean distance were also examined in certain other classes of graphs [18], [13]. Chung [7] showed that the independence number is an upper bound the mean distance and Dankelmann [9] proved upper bounds for it in terms of the so called k-packing number. Average eccentricty (a distance related notion), was recently studied in [9].…”

Section: Introductionmentioning

confidence: 99%