This paper studies generalized variants of the MAXIMUM INDEPENDENT SET problem, called the MAXIMUM DISTANCE-d INDEPENDENT SET problem (MaxDdIS for short). For an integer d≥2, a distance-d independent set of an unweighted graph G=(V, E) is a subset S⊆V of vertices such that for any pair of vertices u, v∈S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r≥3) and planar graphs, as follows: (1) For every fixed integers d≥3 and r≥3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d=r=3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

Let G be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. Suppose that we are given two list edge-colorings f0 and fr of G, and asked whether there exists a sequence of list edge-colorings of G between f0 and fr such that each list edge-coloring can be obtained from the previous one by changing a color assignment of exactly one edge. This problem is known to be PSPACE-complete for every integer k ≥ 6 and planar graphs of maximum degree three, but any computational hardness was unknown for the non-list variant in which every edge has the same list of k colors. In this paper, we first improve the known result by proving that, for every integer k ≥ 4, the problem remains PSPACE-complete even for planar graphs of bounded bandwidth and maximum degree three. Since the problem is known to be solvable in polynomial time if k ≤ 3, our result gives a sharp analysis of the complexity status with respect to the number k of colors. We then give the first computational hardness result for the non-list variant: for every integer k ≥ 5, the non-list variant is PSPACE-complete even for planar graphs of bandwidth quadratic in k and maximum degree k.

Given a graph G=(V, E), where V and E are vertex and edge sets of G, and a subset VNT of vertices called a non-terminal set, the minimum spanning tree with a non-terminal set VNT, denoted by MSTNT, is a connected and acyclic spanning subgraph of G that contains all vertices of V with the minimum weight where each vertex in a non-terminal set is not a leaf. On general graphs, the problem of finding an MSTNT of G is NP-hard. We show that if G is an outerplanar graph then finding an MSTNT of G is linearly solvable with respect to the number of vertices.

For k ≥ 3, a convex geometric graph is called k-locally outerplanar if no path of length k intersects itself. In [D. Boutin, Convex Geometric Graphs with No Short Self-intersecting Path, Congressus Numerantium 160 (2003) 205-214], Boutin stated the results of the degeneracy for 3-locally outerplanar graphs. Later, in [D. Boutin, Structure and Properties of Locally Outerplanar Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 60 (2007) 169-180], a structural property on k-locally outerplanar graphs was proposed. These results are based on the existence of “minimal corner pairs”. In this paper, we show that a “minimal corner pair” may not exist and give a counterexample to disprove the structural property. Furthermore, we generalize the result on the degeneracy with respect to k-locally outerplanar graphs.

A queue layout of a graph G consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The queuenumber qn(G) is the minimum number of queues required in a queue layout of G. The Cartesian product of two graphs G1 = (V1,E1) and G2 = (V2,E2), denoted by G1 × G2, is the graph with {:v1 ∈ V1 and v2 ∈ V2} as its vertex set and an edge (,) belongs to G1×G2 if and only if either (u1,v1) ∈ E1 and u2 = v2 or (u2,v2) ∈ E2 and u1 = v1. Let Tk1,k2,...,kn denote the n-dimensional toroidal grid defined by the Cartesian product of n cycles with varied lengths, i.e., Tk1,k2,...,kn = Ck1 × Ck2 × … × Ckn, where Cki is a cycle of length ki ≥ 3. If k1 = k2 = … = kn = k, the graph is also called the k-ary n-cube and is denoted by Qnk. In this paper, we deal with queue layouts of toroidal grids and show the following bound: qn(Tk1,k2,...,kn) ≤ 2n-2 if n ≥ 2 and ki ≥ 3 for all i = 1,2,...,n. In particular, for n = 2 and k1,k2 ≥ 3, we acquire qn(Tk1,k2) = 2. Recently, Pai et al. (Inform. Process. Lett. 110 (2009) pp.50-56) showed that qn(Qnk) ≤ 2n-1 if n ≥1 and k ≥9. Thus, our result improves the bound of qn(Qnk) when n ≥2 and k ≥9.

Let G be a connected graph in which we designate a vertex or a block (a biconnected component) as the center of G. For each cut-vertex v, let Gv be the connected subgraph induced from G by v and the vertices that will be separated from the center by removal of v, where v is designated as the root of Gv. We consider the set R of all such rooted subgraphs in G, and assign an integer, called an index, to each of the subgraphs so that two rooted subgraphs in R receive the same indices if and only if they are isomorphic under the constraint that their roots correspond each other. In this paper, assuming a procedure for computing a signature of each graph in a class of biconnected graphs, we present a framework for computing indices to all rooted subgraphs of a graph G with a center which is composed of biconnected components from . With this framework, we can find indices to all rooted subgraphs of a outerplanar graph with a center in linear time and space.

In a rooted triangulated planar graph, an outer vertex and two outer edges incident to it are designated as its root, respectively. Two plane embeddings of rooted triangulated planar graphs are defined to be equivalent if they admit an isomorphism such that the designated roots correspond to each other. Given a positive integer n, we give an O(n)-space and O(1)-time delay algorithm that generates all biconnected rooted triangulated planar graphs with at most n vertices without delivering two reflectively symmetric copies.

In a rooted graph, a vertex is designated as its root. An outerplanar graph is represented by a plane embedding such that all vertices appear along its outer boundary. Two different plane embeddings of a rooted outerplanar graphs are called symmetric copies. Given integers n ≥ 3 and g ≥ 3, we give an O(n)-space and O(1)-time delay algorithm that generates all biconnected rooted outerplanar graphs with exactly n vertices such that the size of each inner face is at most g without delivering two symmetric copies of the same graph.

The planar Hajos calculus (PHC) is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajos calculus (PHC(dd)) is PHC with the restriction that all the graphs that appear in the construction (including a final graph) must have maximum degree at most d. We prove the followings: (1) If PHC is polynomially bounded, then for any d ≥ 4, PHC(dd+2) can generate any non-3-colorable planar graphs of maximum degree at most d in polynomial steps. (2) If PHC can generate any non-3-colorable planar graphs of maximum degree 4 in polynomial steps, then PHC is polynomially bounded.

The Hajos calculus is a nondeterministic procedure which generates the class of non-3-colorable graphs. If all non-3-colorable graphs can be constructed in polynomial steps by the calculus, then NP = co-NP holds. Up to date, however, it remains open whether there exists a family of graphs that cannot be generated in polynomial steps. To attack this problem, we propose two graph calculi PHC and PHC* that generate non-3-colorable planar graphs, where intermediate graphs in the calculi are also restricted to be planar. Then we prove that PHC and PHC* are sound and complete. We also show that PHC* can polynomially simulate PHC.

The minimum vertex ranking spanning tree problem is to find a spanning tree of G whose vertex ranking is minimum. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of interval graphs. This paper proposes a polynomial time algorithm for solving the minimum vertex ranking spanning tree problem on outerplanar graphs.

In this paper we show that an outerplanar graph G with maximum degree at most 3 has a 2-D orthogonal drawing with no bends if and only if G contains no triangles. We also show that an outerplanar graph G with maximum degree at most 6 has a 3-D orthogonal drawing with no bends if and only if G contains no triangles.

A plane graph is a planar graph with a fixed embedding. In a no-bend orthogonal drawing of a plane graph, each vertex is drawn as a point and each edge is drawn as a single horizontal or vertical line segment. A planar graph is said to have a no-bend orthogonal drawing if at least one of its plane embeddings has a no-bend orthogonal drawing. In this paper we consider a class of planar graphs, called subdivisions of planar triconnected cubic graphs, and give a linear-time algorithm to examine whether such a planar graph G has a no-bend orthogonal drawing and to find one if G has.

Given an edge-weighted graph G, the minimum maximal matching problem asks to find a minimum weight maximal matching. The problem is known to be NP-hard even if the graph is planar and unweighted. In this paper, we consider the problem in planar graphs. First, we prove a strong inapproximability for the problem in weighted planar graphs. Second, in contrast with the first result, we show that a polynomial time approximation scheme (PTAS) for the problem in unweighted planar graphs can be obtained by a divide-and-conquer method based on the planar separator theorem. For a given ε > 0, our scheme delivers in time a solution with size at most (1 + ε) times the optimal value, where n is the number of vertices in G and α is a constant number.

This paper analyzes what structural features of graph problems allow efficient parallel algorithms. We survey some parallel algorithms for typical problems on three kinds of graphs, outerplanar graphs, trapezoid graphs and in-tournament graphs. Our results on the shortest path problem, the longest path problem and the maximum flow problem on outerplanar graphs, the minimum-weight connected dominating set problem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs are adopted as working examples.

Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.

Let W be a real symmetric matrix associated with a weighted 2-connected planar graph. It is important to study a fast algorithm to solve the linear system Wx = c, since the system has many various applicaions, for example to solve partial defferencial equations numerically. In this paper, a new algorithm for the solution of a linear system of equations by Δ-Y transformations is proposed, and a sufficient condition for using this algorithm is proved. We show that this algorithm solves in O (n3/2) time a linear system associated with a planar graph which is embedded a cylinder graph with n vertices.

We define a new framework for rewriting graphs, called a formal graph system (FGS), which is a logic program having hypergraphs instead of terms in first-order logic. We first prove that a class of graphs is generated by a hyperedge replacement grammar if and only if it is defined by an FGS of a special form called a regular FGS. In the same way as logic programs, we can define a refutation tree for an FGS. The classes of TTSP graphs and outerplanar graphs are definable by regular FGSs. Then, we consider the problem of constructing a refutation tree of a graph for these FGSs. For the FGS defining TTSP graphs, we present a refutation tree algorithm of O(log2nlogm) time with O(nm) processors on an EREW PRAM. For the FGS defining outerplanar graphs, we show that the refutation tree problem can be solved in O(log2n) time with O(nm) processors on an EREW PRAM. Here, n and m are the numbers of vertices and edges of an input graph, respectively.