Delaunay and Gabriel graphs are widely studied geometric proximity structures. Motivated by applications in wireless routing, relaxed versions of these graphs known as \emph{Locally Delaunay Graphs} ($LDGs$) and \emph{Locally Gabriel Graphs} ($LGGs$) were proposed. We propose another generalization of $LGGs$ called \emph{Generalized Locally Gabriel Graphs} ($GLGGs$) in the context when certain edges are forbidden in the graph. Unlike a Gabriel Graph, there is no unique $LGG$ or $GLGG$ for a given point set because no edge is necessarily included or excluded. This property allows us to choose an $LGG/GLGG$ that optimizes a parameter of interest in the graph. We show that computing an edge maximum $GLGG$ for a given problem instance is NP-hard and also APX-hard. We also show that computing an $LGG$ on a given point set with dilation $\le k$ is NP-hard. Finally, we give an algorithm to verify whether a given geometric graph $G=(V,E)$ is a valid $LGG$.
A geometric graph G = (V, E) is an embedding of the set V as points in the plane and the set E as line segments joining two points in V . Delaunay graphs, Gabriel graphs and Relative neighborhood graphs (RNGs) are classic examples of geometric graphs that have been extensively studied and have applications in computer graphics, GIS, wireless networks, sensor networks, etc (see survey [8]). Gabriel and Sokal [6] defined the Gabriel graph as follows: Definition 1. A geometric graph G = (V, E) is called a Gabriel graph if the following condition holds: For any u, v ∈ V , an edge (u, v) ∈ E if and only if the circle with uv as diameter does not contain any other point of V .
Gabriel graphs have been used to model the topology in a wireless network [3,12]. Motivated by applications in wireless routing, Kapoor and Li [9] proposed a relaxed version of Delaunay/Gabriel graphs known as k-locally Delaunay/Gabriel graphs. The edge complexity of these structures has been studied in [9,11]. In this paper, we focus on 1-locally Gabriel graphs and call them Locally Gabriel Graphs (LGGs). Definition 2. A geometric graph G = (V, E) is called a Locally Gabriel Graph if for every (u, v) ∈ E, the circle with uv as diameter does not contain any neighbor of u or v in G.
The above definition implies that in an LGG, two edges (u, v) ∈ E and (u, w) ∈ E conflict with each other and cannot co-exist if ∠uwv ≥ π 2 or ∠uvw ≥ π 2 . Conversely if edges (u, v) and (u, w) co-exist in an LGG, then ∠uwv < π 2 and ∠uvw < π 2 . We call this condition as LGG constraint. Study of these graphs was initially motivated by design of dynamic routing protocols for ad hoc wireless networks [10]. An ad-hoc wireless network consists of a collection of wireless transceivers (corresponds to the the points) and an underlying network topology (corresponds to the edges) that is used for communication. Like Gabriel Graphs, LGGs are also proximity based structures that capture the interference patterns in wireless networks. An interesting point to be noted is that there is no unique LGG on a given point set since no edge in an LGG is necessarily included or excluded. Thus the edge set of the graph (used for wireless communication) can be customized to optimize certain network parameters depending on the application. While a Gabriel graph has linear number of edges (planar graph), an LGG can be constructed with super-linear number of edges [5]. A dense network can be desirable for applications like broadcasting or multicasting where a large number of pairs of nodes need to communicate with each other and links have limited bandwidth. The dilation or spanning ratio of a graph is an important parameter in wireless network design. It is the maximum ratio of the distance in the network (length of the shortest path) to the Euclidean distance for any two nodes in a wireless network. Graphs with small spanning ratios are important in many applications and motivate the study of geometric spanners (refer to [4] for a survey). Proximity graphs have been studied for their dilation. Some interesting bounds for the dilation of Gabriel Graphs were presented in [2]. In this paper, we initiate study for dilation on LGGs. We show that for certain point sets there exist LGGs with O(1) dilation whereas the Gabriel graph on the same point set has dilation Ω( √ n). In many situations, certain links are forbidden in a network due to physical barriers, visibility constraints or limited transmission radius. Thus, all proximate pairs of nodes might not induce edges and this effect can be considered in LGGs. Thus, it is natural to study LGGs in the context when the network has to be built only with a set of predefined links. In this context, we define a generalized version of LGGs called Generalized locally Gabriel Graphs (GLGGs). Edges in a GLGG can be picked only from the edges in a given predefined geometric graph.
LGGs have focused on obtaining combinatorial bounds on the maximum edge complexity. In [9], it was shown that an LGG has at most O(n 3 2 ) edges since K 2,3 is a forbidden subgraph. Also, it was observed in [11] that any unit distance graph is also a valid LGG. Hence there exist LGGs with Ω(n 1+ c log log n ) edges [5]. It is not known whether an edge maximum LGG can be computed in polynomial time.
Our Contribution: We present the following results in this paper.
- We show that computing a GLGG with at least m edges on a given geometric graph G = (V, E) is NP-complete (reduction from 3-SAT) and also APXhard (reduction from MAX-(3,4)-SAT). 2. We show that the problem of determining whether there exists an LGG with dilation ≤ k is NP-hard by reduction from the partition problem motivated by [7]. We also show that there exists a point set P such that any LGG on P has dilation Ω( √ n) that matches with the best known upper bound [2]. 3. For a given geometric graph G = (V, E), we give an algorithm with running time
2 Hardness of computing an edge maximum GLGG
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