Dear Referee,

Many thanks for your attention to our paper and useful remarks.
Almost all your suggestions are accepted, please see below.

All the best,
Anna Ivanova



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The referees report on the paper
''The weight of 3-paths in sparse planar graphs''
by V.A. Aksenov, O.V. Borodin and A.O. Ivanova

The paper deals with the structure of 3-vertex paths in plane graphs of minimum
degree at least 2 with prescribed girth. This topic has been recently intensively studied by
the above authors as well as by researchers from Kosice graph theory school. As main
results, the authors prove that each plane graph of minimum degree at least 2 and girth 6
contains either a 3-path of weight at most 9, or a 4-path consisting of three 2-vertices an one
inner vertex with potentially unbounded degree; for the girth between 7 and 9, the
existence of 3-path of weight at most 9 is proved. These results improve the very recent
results of Jendro> and Macekova and, also, establish the best upper bounds for several cases
which were left open in their previous work.

The obtained results are new and interesting, therefore, I do recommend the paper
to be published in Electronic Journal of Combinatorics after incorporating several
suggestions which concern mainly an improvement of the paper presentation:

p. 1, l. 5: girth, g, --> girth g
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Thanks, done.

p. 1, l. 15: that is the number --> that is, the number
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Thanks, done.

p. 2, l. 7: In order to describe better various facets of the problematics of edge weight,
I would include (after \delta\ge2) a bridging sentence like Anyway, its finiteness may be enforced
by certain additional constraints based, for example, on degree properties of particular
subgraphs. For example, ...
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Many thanks, done.

p. 2, l. -18: this construction deserves to be described in more details (namely, the symbol
K*_{2,2t} refers, in my opinion, to a configuration which contains a t-vertex path in some way,
thus, ``two adjacent 3-vertices lying in two common 3-faces'' should refer to K*_{2,4})
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Many thanks, done as follows:

$K^*_{2,2t}$ ``two adjacent 3-vertices lying in two common 3-faces'' -->
$K^*_{2,2t}$ obtained from the double $2t$-pyramid by deleting a $t$-matching from the $2t$-cycle
formed by 4-vertices

p. 2, l.-1: y_y --> y_i
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Thanks, done.

p. 4, Figure 1: It would help a reader to assign some labels to half-edges and the same labels
to interior edges incident with the bounding cycle to better stress the way how two halfs of
the graph are connected
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Well, we decided to refrain from labels but rather give this verbal explanation
at the end of Section 2:

More specifically, the bounding cycle of the graph to be obtained
may be encoded as $5,3,5,3,\ldots$ according
to the degrees of its vertices. Moreover, its internal half may be encoded as
$5_2,3_1,5_1,3_0\ldots$, where the subscripts show the number of ingoing edges.
For the exterior half, we have a similar encoding
$5_1,3_0,5_2,3_1\ldots$, so the two halves can be glued in this order.

p. 5, l. 3: So --> Thus,
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Thanks, done.

p. 8, l. -5: 101-103 --> 101-113 [this was checked from the original source available at
http://dml.cz/bitstream/handle/10338.dmlcz/126508/MathSlov_05-1955-2_5.pdf; note that
the original paper is written in Slovak, with Russian summary]
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Many thanks, done.

p. 8, l. -16: Macekova Describing --> Macekova, Describing
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Thanks, done.
