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             THE ELECTRONIC JOURNAL OF COMBINATORICS

         REPORT FORM AND SCORECARD FOR EVALUATING PAPERS

TITLE OF PAPER:  Lower bounds on the obstacle number of graphs.

AUTHOR OF PAPER: Padmini Mukkamala, Janos Pach and Domotor Palvolgyi

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REPORT:

This paper proves elegant bounds on the maximum obstacle number of n-vertex graphs. These results will be of interest in the combinatorial geometry and graph drawing communities. The paper is very well written. I recommend that it be accepted.

Some comments, mostly minor, follow:

The discussion about the obstacle number of \overline{K_n} when the obstacles must be points reads as if O(n) is the conjectured answer. However, everyone thinks that the answer is superlinear.

DONE

Regarding the upper bound of n2^{O(sqrt(log n))} discussed on page 2, the same proof technique was independently used for a very similar question by Yonutz V. Stanchescu in "Planar sets containing no three collinear points and non-averaging sets of integers", Discrete Mathematics  256(1-2):387–395, 2002.

DONE

Add the following reference to the discussion about the obstacle number of \overline{K_n} when the obstacles must be points on page 2:  A. Pór and D. R. Wood. On visibility and blockers, J. Computational Geometry 1:29-40, 2010. Say that this question is commonly called "the blocking conjecture".

DONE

Explain why you divide by h! near the end of the proof of Claim 4. Also, I would call this "Claim" a "Lemma".

h! REMOVED, NAME KEPT AS CLAIM

The proof of Theorem 2 is clever. Reducing the question to these disjoint subgraphs is a nice idea.

THANK YOU!

Reference [16] has appeared in Graphs and Combinatorics 27 (2011), 465-473.

DONE

Add all the details for reference [11]:  Graphs with large obstacle numbers, Proc. WG 2010: 36th Intern. Workshop on Graph Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 6410, Springer-Verlag, Berlin, 2010, 292-303.

DONE

Replace "h" by "$h$" in reference [9].

WOW, good eyes...

References [1] and [6] are in: Thirty Essays in Geometric Graph Theory, Ed. J. Pach, Algorithms & Combinatorics Series, Springer, 2012.

DONE

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(A) The importance of the research in this paper to the field in which that research lies is

2. intermediate.


(B) This field of research is

2. somewhat important.
