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The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2003)

ESAIM: Control, Optimisation and Calculus of Variations

In [30], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation...

The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem...

The triangles method to build X -trees from incomplete distance matrices

Alain Guénoche, Bruno Leclerc (2001)

RAIRO - Operations Research - Recherche Opérationnelle

A method to infer X -trees (valued trees having X as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2 n -3 distance values between the n elements of X , if they fulfill some explicit conditions. This construction is based on the mapping between X -tree and a weighted generalized 2-tree spanning X .

The triangles method to build X-trees from incomplete distance matrices

Alain Guénoche, Bruno Leclerc (2010)

RAIRO - Operations Research

A method to infer X-trees (valued trees having X as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2n-3 distance values between the n elements of X, if they fulfill some explicit conditions. This construction is based on the mapping between X-tree and a weighted generalized 2-tree spanning X.

The Turán Number of the Graph 2P5

Halina Bielak, Sebastian Kieliszek (2016)

Discussiones Mathematicae Graph Theory

We give the Turán number ex (n, 2P5) for all positive integers n, improving one of the results of Bushaw and Kettle [Turán numbers of multiple paths and equibipartite forests, Combininatorics, Probability and Computing, 20 (2011) 837-853]. In particular we prove that ex (n, 2P5) = 3n−5 for n ≥ 18.

The Turàn number of the graph 3P4

Halina Bielak, Sebastian Kieliszek (2014)

Annales UMCS, Mathematica

Let ex (n,G) denote the maximum number of edges in a graph on n vertices which does not contain G as a subgraph. Let Pi denote a path consisting of i vertices and let mPi denote m disjoint copies of Pi. In this paper we count ex(n, 3P4)

The upper domination Ramsey number u(4,4)

Tomasz Dzido, Renata Zakrzewska (2006)

Discussiones Mathematicae Graph Theory

The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kₚ with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kₚ induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4,4) ≤ 15.

The upper traceable number of a graph

Futaba Okamoto, Ping Zhang, Varaporn Saenpholphat (2008)

Czechoslovak Mathematical Journal

For a nontrivial connected graph G of order n and a linear ordering s v 1 , v 2 , ... , v n of vertices of G , define d ( s ) = ∑ i = 1 n - 1 d ( v i , v i + 1 ) . The traceable number t ( G ) of a graph G is t ( G ) = min { d ( s ) } and the upper traceable number t + ( G ) of G is t + ( G ) = max { d ( s ) } , where the minimum and maximum are taken over all linear orderings s of vertices of G . We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t + ( G ) - t ( G ) = 1 are characterized and a formula for the upper...

The use of Euler's formula in (3,1)*-list coloring

Yongqiang Zhao, Wenjie He (2006)

Discussiones Mathematicae Graph Theory

A graph G is called (k,d)*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ∈ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. Ko-Wei Lih et al. used the way of discharging to prove that every planar graph without 4-cycles and i-cycles for some i ∈ 5,6,7 is (3,1)*-choosable. In this paper, we show that if G is 2-connected, we may just use Euler’s formula and the graph’s structural properties to prove...

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