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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 (expand as necessary):

Comments on cited references:

(1)  When I looked at the reference for Theorem B, I saw that that paper
states, "We assume, without loss of generality, that the face of interest
in A(S) is the face at infinity," and then goes on to prove the theorem for
only the face at infinity.  I asked the authors about this, and they came
up with the following explanation:

"Let me try a simple argument to convince you that the face at infinity
result is in fact without loss of generality:

Take a face f that is not the face at infinity.
It is multiply connected -- it has an outer boundary, and various holes.
The outer boundary is partitioned by its leftmost/rightmost points
into two simple chains, an 'upper' and a 'lower'.
Some segments contribute to the upper, some to the lower boundary, but no
straight line segment contributes to both.
Thus, partition the set of segments contributing to the outer boundary
into
red/blue according to contributions to upper/lower outer boundary.
There are also segments ('black') that contribute to holes of f, and
segments ('green') that lie exterior to f and do not contribute
to the outer boundary or to the holes of f.

Now, apply our result for the face at infinity in the arrangement of
red and black segments, and then separately the blue and black
segments.  Combine to get f."

In your paper you may want to paraphrase this argument and note that this
part of the proof is missing from the original paper.

I THINK THIS ARGUMENT, WHILE CERTAINLY INTERESTING, IS QUITE FAR FROM THE MAIN FOCUS OF OUR PAPER, AND SINCE THE CITED THEOREM IS OLD AND WELL-KNOWN, THERE IS NO NEED TO INCLUDE THIS PROOF.

(2)  At the end of section 1, you write, "It is an interesting open
problem to decide whether the obstacle number of planar graphs can be
bounded from above by a constant.  For outerplanar graphs, this has been
verified by Fulek, Saeedi, and Sarioz [6], who proved that every
outerplanar graph has obstacle number at most 5."  The problem of planar
graphs was posed in your reference [2], which also proves that every
outerplanar graph has obstacle number (at most) 1 (Theorem 5).  Your
reference [6] proves the bound of 5 for convex obstacle number on
outerplanar graphs.  (Note that you do not define convex obstacle number
until later on the same page.)

DONE

(3)  Below the statement of Theorem 2, you write, "This comes close to
answering the question in [2] whether there exist graphs with n vertices
and obstacle number at least n."  But I do not think this question appears
in reference [2].

CHANGED

Comments on clarity:

(4)  I was not familiar with the definition of big-O notation in multiple
variables.  Perhaps it would be helpful to include a reference such as "On
Asymptotic Notation with Multiple Variables", by Rodney R. Howell.  Also,
because I am not accustomed to working with asymptotics, I would prefer to
see log^2 n written as (log n)^2 (at least the first time) so that I know
it isn't log log n.

I THINK THESE ARE QUITE STANDARD NOTATIONS, SO LEFT AS WAS.
THE DEFINION OF big-O DOES NOT DEPEND OF THE VARIABLE INSIDE IT. IT MEANS THERE IS C SUCH THAT THE LHS IS AT MOST C TIME THE RHS.

(5)  Below the statement of Theorem 1, you write, "In the above bounds, it
makes no difference whether we count labeled or unlabeled graphs, because
the number of labeled graphs is at most n! = 2^{O(n log n)} times the
number of unlabeled ones."  Can you state precisely what you mean by "it
makes no difference"?  I thought you might mean that counting unlabeled
graphs would not give a stronger bound, but I didn't think this was proven
by the second half of your statement.

IT MEANS THAT IF THE THEOREM IS TRUE FOR UNLABELED GRAPHS, IT ALSO HOLDS FOR LABELED ONES.

(6)  I like the proof of Claim 4, but I don't think it helps explain the
proof of Theorem 1.  Maybe it would go better in section 3.

IT WOULD BE ALSO STRANGE TO INCLUDE IT IN SECTION 3, SO MAYBE WE WOULD JUST KEEP IT WHERE IT IS, AS IT MIGHT GIVE A GOOD WARM UP FOR THE LATER PROOFS.

(7)  In the proof of Theorem 1, at the end of the paragraph proving that
(ii)=>(i), I found it helpful to assume h is at most n choose 2, in the log
s term, and I thought you may want to say this.

DONE

(8)  In the proof of Theorem 2, you cite Theorem 1(i) substituting h = 1.
Depending on the definition of big-O notation with multiple variables, I am
not sure this is valid, but I found it almost as easy to use Theorem 1(ii)
substituting s = O(k log k).

THAT WOULD BE ANOTHER OPTION, BUT I THINK FOR h=1 THERE CAN BE NO CONFUSION ABOUT THE MEANING OF big-O.
