1 Introduction
A sequence of points in the Euclidean plane , , …, is called monotone if the sequence of their coordinates, i.e. , , …, , is either decreasing or increasing, with denoting the coordinate of the point . A geometric path , , …, is called monotone if the sequence of its vertices, i.e. the sequence , , …, , is monotone. If is monotone for some axis then is called monotone. Let be a geometric graph. If each are connected by a monotone path then is called monotone. If is monotone for some axis then is called uniform monotone (following the terminology of [22]). Uniform monotone graphs were called monotone graphs by Angelini [3]. If each are connected by a monotone path, where the direction of monotonicity might differ for different pairs of vertices, then is called monotone. Monotone graphs were introduced by Angelini et al. [4]. Drawing an (abstract) graph as a monotone (geometric) graph has been a topic of research [3, 4, 5, 13, 24].
The Monotone Minimum Spanning Graph problem, i.e. the problem of constructing the monotone spanning geometric graph of a given point set that has the minimum cost, where the cost of a geometric graph is the sum of the Euclidean lengths of its edges, was recently introduced (but not solved) in [22] and it remains an open problem whether it is NPhard. Since the more general (without the requirement of monotonicity) Euclidean Minimum Spanning Tree problem can be solved in time [27], this constitutes a great differentiation that is induced by the addition of the property of monotonicity.
A point set is rooted if there exist points , , …, distinguished from the other points of which are called the roots of . A geometric graph is called rooted if is rooted and its roots are the roots of . A rooted geometric graph is rooted monotone if each root and each point are connected by monotone paths. Similarly, is rooted uniform monotone (following the terminology of [22]) if it is rooted monotone for some axis . For simplicity, we may also denote point sets or geometric graphs that are rooted simply as rooted. A polygon that is rooted monotone, in which its roots are its lowest and highest vertices, can be triangulated in linear time [11]. Lee and Preparata [16] preprocessed a subdivision of the plane such that the region in which a query point belongs can be found quickly, by (i) extending the geometric graph bounding to a rooted monotone planar geometric graph in which the roots are the highest and lowest vertices of , and (ii) constructing a set of appropriate monotone paths from the lowest to the highest vertex of . Additionally, Lee and Preparata [16] noted that a rooted planar geometric graph, where all vertices have different coordinates, in which the roots are the highest and lowest vertices of the graph is rooted monotone if and only if each nonroot vertex has both a neighbor above it and a neighbor below it. Furthermore, a rooted geometric graph , where all vertices have different coordinates, with a (single) root that is not the highest or lowest point of is rooted monotone if and only if each nonroot vertex has a neighbor such that is between (inclusive) and [22]. Additionally, rooted uniform monotone graphs can be efficiently recognized [22]. The rooted monotone (uniform monotone) minimum spanning graph (following the terminology of [22]) of a rooted point set is the rooted monotone (uniform monotone) spanning graph of that has the minimum cost. The rooted monotone (uniform monotone) minimum spanning graph^{2}^{2}2In [22] it is shown that it is actually a tree. of a rooted point set can be produced in (resp., ) time [22]. The problem of drawing a rooted tree as a rooted monotone minimum spanning graph is studied in [20]. The (rooted) monotone minimum spanning graph of a point set is the geometric path that traverses all the points of by moving north, from the lowest point to the highest point of [22]. Regarding the problem of producing the rooted monotone minimum spanning graph of a rooted point set , with , it is an open problem, posed in [22], whether it is NPhard.
The restricted fathers tree problem was introduced in [12] and is related to the rooted monotone minimum spanning graph problem constrained to rooted point sets in which the coordinate of the root is zero and the coordinates of the other points of are all negative (or all positive). The input of the restricted fathers tree problem is a complete graph with root where each edge has a cost and each vertex has a value and the goal is to output the spanning tree in which the path from the root to each vertex decreases in value that has the minimum cost. The restricted fathers tree problem is greedily solvable [12, Corollary 2.6].
A geometric path , , …, is monotone if the sequence of its vertices is both monotone, i.e. the sequence , , …, , is monotone, and monotone. is 2Dmonotone (following the terminology of [22]) if it is monotone for some orthogonal axes . A geometric graph is 2Dmonotone (following the terminology of [22]) if each pair of points of is connected by a 2Dmonotone path. 2Dmonotone paths/graphs were called anglemonotone paths/graphs by Bonichon et al. [8]. Bonichon et al. [8] showed that deciding if a geometric graph is 2Dmonotone can be done in time. Triangulations with no obtuse internal angles are 2Dmonotone graphs [10, 19]. There exist point sets for which any 2Dmonotone spanning graph is not planar [8]. The problem of constructing 2Dmonotone graphs with asymptotically less than quadratic edges was studied by Lubiw and Mondal [18]. It is an open problem, posed in [22], whether the 2Dmonotone spanning graph of a point set that has the minimum cost can be efficiently computed.
The (rooted) monotone and (rooted) uniform 2Dmonotone (using the terminology of [22]) graphs are defined similar to the (rooted) monotone and (rooted) uniform monotone graphs. Deciding if a rooted geometric graph is rooted monotone (uniform 2Dmonotone) can be done in (resp., ) time [22]. Additionally, the rooted monotone (uniform 2Dmonotone) spanning graph of a rooted point set that has the minimum cost^{3}^{3}3In [22] it is shown that it is actually a tree, denoted as the rooted monotone (uniform 2Dmonotone) minimum spanning tree in [22] and abbreviated as the rooted MMST (resp., rooted UMMST) in [22]. can be computed in (resp., ) time [22]. We focus on the production of the monotone minimum spanning graph (MMSG) of a point set , i.e. the monotone spanning graph of that has the minimum cost, and the production of the uniform monotone minimum spanning graph (UMMSG) of a point set , i.e. the uniform monotone spanning graph of that has the minimum cost. We also study the corresponding problems regarding the production of the spanning graphs with the least number of edges, i.e. the production of the monotone spanning graph with the least number of edges and the production of the uniform monotone spanning graph with the least number of edges.
A curve is increasingchord [15, 26] if for each traversed in this order along it, the length of the line segment is greater than or equal to the length of . Alamdari et al. [1] introduced increasingchord graphs which are the geometric graphs for which each two vertices are connected by an increasingchord path. Increasingchord graphs are widely studied [1, 6, 10, 21, 23]. The problem of producing increasingchord spanning graphs (where Steiner points may be added) of a point set was studied in [1, 10, 21]. The approach employed in [1, 10, 21], was to connect the points of by 2Dmonotone paths since as noted by Alamdari et al. [1] 2Dmonotone paths are also increasingchord paths.
Let be a point set and let then and are rectangularly visible if the rectangle with corners and does not include any other point of . Furthermore, the rectangle of influence graph of is the geometric graph spanning such that is an edge of the graph if and only if and are rectangularly visible. Alon et al. [2] denoted rectangularly visible points as separated points and the rectangle of influence graph as the separation graph. Computing the rectangle of influence graph of can be done in time [25]. There exist point sets for which the number of edges of their rectangle of influence graph is [2]. The rectangle of influence graph does not remain the same if the Cartesian System is rotated [14, Proposition 3]. Drawing an abstract graph as a rectangle of influence graph has been studied [17].
Our Contribution. Building upon previous results, we easily obtain that given a point set the MMSG of is equal to the monotone spanning graph of that has the least number of edges and are both equal to the rectangle of influence graph of . We note that given a point set the UMMSG of does not necessarily coincide with the uniform 2Dmonotone spanning graph of that has the least number of edges. We also show that both the UMMSG of and the uniform 2Dmonotone spanning graph of that has the least number of edges can be produced in time. Additionally, we give a simple approximation algorithm for the problem of producing the rooted monotone minimum spanning graph of a rooted point set.
2 Preliminaries
2.1 Monotone Minimum Spanning Graphs
Angelini [3] noted the following Fact regarding monotone graphs.
Fact 1 (Angelini [3]).
Let be a monotone graph where no two points of have the same coordinate and let such that for each the sequence is not monotone. Then, and are adjacent in .
Fact 1 is easily extended in the context of monotone graphs. More specifically, let be a monotone graph and such that for each the sequence of points is not monotone, then and are adjacent in . Alon et al. [2] noted that the points of a point set are rectangularly visible if and only if for each the sequence of points is not monotone. Hence, the rectangle of influence graph of is a subgraph of .
Liotta et al. [17, Lemma 2.1] showed that the rectangle of influence graph of a point set is a monotone graph^{4}^{4}4Technically speaking, Liotta et al. [17] showed that the rectangle of influence graph of a point set is a graph such that each two vertices are connected by a path lying inside the rectangle defined by these vertices but upon careful reading the path that is obtained in their proof is monotone..
From the previous two sentences, regarding the rectangle of influence graph, we obtain the following Corollary.
Corollary 1.
Let be a point set. The MMSG of and the monotone spanning graph of that has the least number of edges coincide and they are both equal to the rectangle of influence graph of .
2.2 Rooted Uniform 2DMonotone Graphs
Mastakas and Symvonis [22] studied the problem of recognizing rooted uniform 2Dmonotone graphs. They initially noted the following Fact.
Fact 2 (Observation 8 in [22]).
Let be a geometric graph with root . If one rotates a Cartesian System , then may become rooted monotone while previously it was not, or vice versa, only when the axis becomes (or leaves the position where it previously was) parallel or orthogonal to

[nosep]

a line passing through and a point .

an edge , where .
Based on Fact 2, Mastakas and Symvonis [22] gave a rotational sweep algorithm denoted as the rooted uniform 2Dmonotone recognition algorithm in [22].
Fact 3 ([22]).
The rooted uniform 2Dmonotone recognition algorithm

[nosep,label=)]

computes, in time, a set of sufficient Cartesian Systems, of size , which are associated with (1) lines passing through and a point and (2) edges , where .

tests, in total time^{5}^{5}5Technically speaking in [22] it is shown that the remaining steps, i.e. the steps after the computation of the sufficient Cartesian Systems, of the rooted uniform 2Dmonotone recognition algorithm take total time. Internally in the rooted uniform 2Dmonotone recognition algorithm given in [22], for each it is stored the set of adjacent points to that are in the rectangle w.r.t. the Cartesian System with corners and , which is denoted as in [22]. Furthermore, it is stored the set of points for which which is denoted as in [22]. However, only the cardinalities of these sets are necessary [22, Lemma 9], hence if instead of the sets and their cardinalities are stored, the remaining steps of the rooted uniform 2Dmonotone recognition algorithm take total time. , if is rooted monotone for some Cartesian System in the previously computed set of sufficient Cartesian Systems.
Fact 4 (Theorem 1 in [22]).
Let be a rooted point set then the rooted monotone minimum spanning graph of can be obtained in time.
3 The 2DUMMSG Problem
We now deal with the construction of the UMMSG and the uniform monotone spanning graph with the least number of edges. We initially show that the UMMSG of a point set can be obtained in time. For this, we employ a rotational sweep technique. Our approach regarding the construction of the UMMSG is similar to the approach employed for the calculation of the rooted uniform monotone spanning graph that has the minimum cost in [22]. We assume that no three points of are collinear and no two line segments and , are parallel or orthogonal.
Let be a point set and be a point of . Let denote the subset of points of that are rectangularly visible from w.r.t. the Cartesian System . See for example, Figure 1(a).
Proposition 2.
If we rotate a Cartesian System counterclockwise, then the MMSG of changes only when reaches or moves away from a line perpendicular or parallel to a line passing through two points of .
Proof.
If we rotate the Cartesian System counterclockwise then the for a point changes only when reaches or moves away from a line perpendicular or parallel to a line passing through two points of ; e.g. see Figure 1. From the previous and since the , equals to the set of adjacent vertices of in the MMSG of (Corollary 1), we obtain the Proposition. ∎
(a)  (b)  (c) 
Let a line of slope is perpendicular or parallel to a line passing through two points of . Let , , …, with such that … . We now define the set to be equal to , , , , …, , . Let , , …, be the Cartesian Systems in which the vertical axis has slope in , ordered w.r.t. the slope of their vertical axis.
Theorem 3.
The uniform monotone minimum spanning graph of a point set can be computed in time.
Proof.
From Proposition 2 and the previous definitions we obtain the following Proposition.
Proposition 4.
The uniform monotone minimum spanning graph of is one of the MMSG of over all Cartesian Systems with of slope in .
We now give a time rotational sweep algorithm. The algorithm initially computes the MMSG of and then it obtains each MMSG of from the MMSG of . Throughout the procedure the Cartesian System in which the algorithm encountered the minimum cost solution so far is stored. In its last step, the algorithm recomputes the MMSG of , which since it is equal to the rectangle of influence graph w.r.t. the Cartesian System (Corollary 1) it can be computed in time [25]. The crucial proposition (which we show later) that makes the time complexity of the algorithm equal to is that each transition from the MMSG of to the MMSG of takes time.
For each two points of let be the number of points of that are included in the rectangle w.r.t. the Cartesian System with opposite vertices and . Then, can be equivalently defined using the quantities , as follows: .
We store the , , ,, …, in the data structure which is implemented as an array of booleans. We also store the , , ,, …, in the variable .
Computing the Cartesian Systems , , , …, can be done in time. Accompanied with each Cartesian System is the pair of points such that is either parallel or perpendicular to the axis or the axis.
Ichino and Sklansky [14] noted that employing a range tree [7, 9] that contains the points of one can calculate i) the rectangle of influence graph of , and ii) the , for a Cartesian System . Applying the previously mentioned approach, noted by Ichino and Sklansky [14], are obtained i) the rectangle of influence graph of w.r.t. the Cartesian System (which by Corollary 1 equals to the MMSG of ), and ii) the , .
We now show that we can update all the , such that from equal to , they become equal to , in total time. For each the update of takes time. This is true, since only the points and have to be tested for inclusion to or removal from . More specifically, we have to test if for one of them, say , the rectangle with corners and contains (or it does not contain) w.r.t. the Cartesian System while it did not contain (or it contained) it w.r.t. . If this is true, then the changes and has to be tested for membership in and included to or removed from . Regarding , the update takes time, since for each other point we have to test if the rectangle with corners and contains (or it does not contain) w.r.t. the Cartesian System while it did not contain it (or it contained it) w.r.t. the and if so update both the and the existence of in if necessary. Similarly, can be updated in time. ∎
We note that the procedure of obtaining the UMMSG can be trivially modified such that the uniform monotone spanning graph of a point set with the least number of edges can be obtained in time. Since for an arbitrary Cartesian System the MMSG of is equal to the monotone spanning graph of with the least number of edges (Corollary 1), the only modification which is necessary is that in the transition from the Cartesian System to the Cartesian System we check if the monotone spanning graph of with the least number of edges has the least number of edges among all the produced solutions so far.
In Figure 2 is given a point set for which the UMMSG of is different from the uniform monotone spanning graph of with the least number of edges.
(a)  (b) 
In Figure 3 we give a point set for which the (nonuniform) monotone spanning graph of with the least number of edges does not coincide to the (nonuniform) monotone spanning graph of that has the minimum cost.
(a)  (b) 
Regarding recognizing uniform monotone graphs, we note that the time rotational sweep algorithm given in [22], which decides if a geometric graph with a specified vertex as root is rooted uniform monotone, can be easily extended into a time rotational sweep algorithm that decides if is uniform monotone. More specifically, in order to decide if is uniform monotone, the rooted geometric graphs , , …, where is the geometric graph with root and , , …, is the vertex set of , are considered. A Cartesian System is rotated counterclockwise. From Fact 2, it follows that one of these rooted geometric graphs becomes rooted monotone while previously it was not, or vice versa, only when the axis becomes (or leaves the position where it was previously) parallel or orthogonal to a line passing through two points of . Hence, Cartesian Systems need to be considered, which can be computed in time. When the becomes (or leaves the position that it previously was) parallel or perpendicular to a line passing through the points then by Fact 2 the status, i.e. being rooted monotone, of the rooted geometric graphs and may change. Hence, the steps of the rooted uniform 2Dmonotone recognition algorithm given in [22] for handling the event associated with the current Cartesian System regarding the rooted geometric graphs and , are applied. Furthermore, if then by Fact 2 it follows that the status, i.e. being rooted monotone, of each , may also change. Hence, for each , the steps of the rooted uniform 2Dmonotone recognition algorithm given in [22] for handling the event associated with the current Cartesian System are applied. Since, the remaining steps, i.e. after the calculation of the sufficient axes, of the rooted uniform 2Dmonotone recognition algorithm, given in [22], regarding any of these rooted geometric graphs take time (Fact 3), applying the remaining steps regarding all these rooted geometric graphs, takes total time.
4 A Approximation Algorithm for the Rooted Monotone Minimum Spanning Graph Problem
We now study the problem of producing the rooted monotone minimum spanning graph of a rooted point set , where . We assume that no two points have the same coordinate.
Let be a point set and then is the subset of points of whose coordinate is greater than . Similarly are defined , and . is the subset of points of whose coordinate is between and . Similarly are defined , and .
In [22, Lemma 1] it is noted that the rooted monotone minimum spanning graph of a rooted point set with root is the union of the rooted monotone minimum spanning graphs of (i) and (ii) . The previous Fact is extended to the following Lemma.
Lemma 5.
Let be a rooted point set, with , where , , …, are the roots of such that … . The rooted monotone minimum spanning graph of is the union of

[nosep]

the rooted monotone minimum spanning graph of .

the rooted monotone minimum spanning graph of .

the rooted monotone minimum spanning graph of , .
Theorem 6.
Given a rooted point set , with , we can obtain in time a rooted monotone spanning graph of with cost at most twice the cost of the rooted monotone minimum spanning graph of .
Proof.
For a rooted point set with roots and that are the lowest and highest points of the point set, respectively, we prove the following Lemma.
Lemma 7.
Given a rooted point set with roots and that are the lowest and highest points of the point set, respectively, we can obtain in time a rooted monotone spanning graph of with cost at most twice the cost of the rooted monotone minimum spanning graph of .
Proof.
Initially, we employ Fact 4 to considering it to have only the root and obtain the geometric graph . Then, we employ Fact 4 to considering it to have only the root , obtaining . In the final step we return the union of and . is rooted monotone since () is rooted monotone with root (resp., ). We now show that has cost at most twice the cost of the rooted monotone minimum spanning graph of . Since, in all the points are connected with () by monotone paths it follows that its cost is greater than or equal to the cost of (resp., ). Hence, the cost of which is less than or equal to the sum of the costs of and is at most twice the cost of . ∎
A rooted planar geometric graph with roots , s.t. , is rooted monotone if and only if for each there exist with [16]. Furthermore, a rooted geometric graph with a (single) root that is not the highest or lowest point of is rooted monotone if and only if for each there exists such that is between (inclusive) and [22]. We extend the previous two Propositions to the following equivalent characterization of rooted monotone graphs where the latter implies an efficient recognition algorithm for rooted monotone graphs.
Proposition 8.
Let be a rooted geometric graph, where , with roots , , …, such that … . is rooted monotone if and only if

[nosep]

for each there exists s.t. .

for each there exists s.t. .

for each there exist s.t. and , , , …, .

there exists s.t. .

there exists s.t. .

there exist s.t. and , .
5 Further Research Directions
Given a point set can the monotone spanning graph of that has the least number of edges be produced in polynomial time?
Does there exist a approximation algorithm, , for the rooted monotone minimum spanning graph problem?
Acknowledgement: I would like to thank Professor Antonios Symvonis for his valuable contribution in developing the results presented in this paper.
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