# Filip Morić's research while affiliated with École Polytechnique Fédérale de Lausanne and other places

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## Publications (15)

Let P be a set of n > d points in ℝd
for d ≥ 2. It was conjectured by Schur that the maximum number of (d − 1)-dimensional regular simplices of edge length diam(P), whose every vertex belongs to P, is n. We prove this statement under the condition that any two of the simplices share at least d − 2 vertices. It is left as an open question to decide...

We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the "host" of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if a...

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to
$\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$
, in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a...

Let P be a set of n points in RdRd. It was conjectured by Schur that the maximum number of (d−1)(d−1)-dimensional regular simplices of edge length diam(P), whose every vertex belongs to P, is n. We prove this statement under the condition that any two of the simplices share at least d−2d−2 vertices. It is left as an open question to decide whether...

Let d1<d2<⋯d1<d2<⋯ denote the set of all distances between two vertices of a convex nn-gon. We show that the number of pairs of vertices at distance d2d2 from one another is at most n+O(1)n+O(1).

Given a set P of n points in \(\mathbb R ^{d}\), let \({d}_{1}>d_{2}>\cdots \) denote all distinct inter-point distances generated by point pairs in \(P\). It was shown by Schur, Martini, Perles, and Kupitz that there is at most one
d-dimensional regular simplex of edge length \({d}_{1}\) whose every vertex belongs to P. We extend this result by sh...

In a seminal paper published in 1946, Erdős initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular par- tial successes and persistent attacks by generations of mathematicians, most problems raised in Erdős’ paper are still unsolved. Given a set of n points in ℝ d , let d 1...

Let S be a set of at least five points in the plane, not all on a line. Suppose that for any three points $${a,b,c\in S}$$ the nine-point center of triangle abc also belongs to S. We show that S must be dense in the plane. We also consider several problems about partitioning the plane into two sets containing their
triangle centers.

In a convex n-gon, let \({d}_{1} > {d}_{2} > \cdots \) denote the set of all distances between pairs of vertices, and let m
i
be the number of pairs of vertices at distance d
i
from one another. Erdős, Lovász, and Vesztergombi conjectured that \(\sum\nolimits_{i\leq k}{m}_{i} \leq kn\). Using a new computational approach, we prove their conjectur...

In a convex n-gon, let d[1] > d[2] > ... denote the set of all distances
between pairs of vertices, and let m[i] be the number of pairs of vertices at
distance d[i] from one another. Erdos, Lovasz, and Vesztergombi conjectured
that m[1] + ... + m[k] <= k*n. Using a new computational approach, we prove
their conjecture when k <= 4 and n is large; we...

We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle. We show
that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5n and 74.2n, respectively. This is a strengthen...

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(vertical bar V vertical bar - 1) - v...

By a poly-line drawing of a graph G on n vertices we understand a drawing of G in the plane such that each edge is represented by a polygonal arc joining its two respective vertices. We call a turning point of a polygonal arc the bend. We consider the class of graphs that admit a poly-line drawing, in which each edge has at most one bend (resp. two...

By a polygonization of a finite point set $S$ in the plane we understand a simple polygon having $S$ as the set of its vertices. Let $B$ and $R$ be sets of blue and red points, respectively, in the plane such that $B\cup R$ is in general position, and the convex hull of $B$ contains $k$ interior blue points and $l$ interior red points. Hurtado et a...

## Citations

... Every graph has a RAC 3 -drawing [12], but not necessarily a RAC 2 -drawing: any graph with a RAC 2 -drawing has at most linearly many edges [4]. The complexity of recognizing graphs with RAC 1 -and RAC 2 -drawings remains tantalizingly open [13,Problem 6]. ...

Reference: RAC-drawability is ER-complete

... In the latter paper the authors also proved that D d (d + 1, n) = 1. In [12] P. Mori´c and J. Pach discussed this conjecture. In particular, they showed that Schur's conjecture holds in the following special case: Theorem 2 (Theorem 2 from [12]). ...

Reference: Proof of Schur's conjecture in $\mathbb R^d$

... Similar questions were considered by Glazyrin and Morić [8], Langi [10], and Pinchasi [11] who studied the perimeter of one or several disjoint polygons covered by a convex body. There is an impressive survey of inequalities concerning the perimeter (and other quantities) of general convex shapes by Scott and Awyong [13]. ...

... Vázsonyi's conjecture was proved independently by B. Grünbaum [10], A. Heppes [11] and S. Straszewicz [20]. An interesting generalization of this result to the case of k-th diameters was obtained by F. Morić and J. Pach [17]. ...

Reference: Proof of Schur’s Conjecture in ℝD

... To focus on the problem discussed in the present paper, we refer to Ismailescu who started to investigate the construction " add the circumcenters (incenters, orthocenters respectively) of all nondegenerate triangles formed by the existing points " . In [4] with Iorio, Radoi˘ ci´c and Silva they proved that in case of the circumcenters the iterative process leads to a dense point set of the plane, and in case of the incenters it always leads to a dense point set in the convex hull of the initial point set. See also [3] for a similar result concerning iterated line intersections. ...

Reference: Incenter iterations in 3-space

... We give a lower bound on the number of edges by using the construction of Arikushi et al. [5] that they used to give the lower bound on edge density for RAC 1 graphs. Let G be Figure 10a. ...

... Ali, Mazorodze, Mukwembi and Vetrík [2] improved Ore's bound by taking into account also the edge-connectivity. Fulek, Morić and Pritchard [16] determined the maximum size of planar graphs of given order and diameter. Bounds on the diameter in terms of order and edge-connectivity were given by Caccetta and Smyth [6]. ...

... It is also shown that if a drawing with orthogonal crossings has at most one or two bends per edge, then its number of edges must be O(n 4 3 ) and O(n 7 4 ), respectively. These last upper bounds were recently improved by Arikushi et al. [2] who showed that if a graph admits a drawing with at most one bend (two bends) per edge and orthogonal crossings, then this graph has at most 6.5n − 13 (74.2n) edges. ...

... In particular covering paths with a small number of edges find applications in robotics and heavy machinery for which turning is an expensive operation [30]. Covering trees with a small number of edges are useful in red-blue separation [20] and in constructing rainbow polygons [19]. In 2010 F. Morić [14] and later Dumitrescu, Gerbner, Keszegh, and Tóth [15] raised many challenging questions about covering paths and trees. ...