Mim-Width I. Induced path problems

The BANDWIDTH minimization problem on graphs of some special graph classes is studied and the following results are obtained. The problem remains NP-complete when restricted to splitgraphs. There is a linear time algorithm to compute the exact bandwidth of a subclass of splitgraphs called hedgehogs. There is an efficient algorithm to approximate the bandwidth of circular permutation graphs within a factor of four.

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n2) or more, and space complexity O(n2) on these graphs. The Hamilton circuit problem is open for this class.We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n2) time and O(n) space algorithm for the Hamilton circuit problem.KeywordsFacility LocationAdjacency RegionPolynomial Time AlgorithmGeometric RepresentationMinimum AngleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Consider a class of graphs $$\mathcal{G}$$ having a polynomial time algorithm computing the set of all minimal separators for every graph in $$\mathcal{G}$$ . We show that there is a polynomial time algorithm for treewidth and minimum fill-in, respectively, when restricted to the class $$\mathcal{G}$$ . Many interesting classes of intersection graphs have a polynomial time algorithm computing all minimal separators, like permutation graphs, circle graphs, circular arc graphs, distance hereditary graphs, chordal bipartite graphs etc. Our result generalizes earlier results for the treewidth and minimum fill-in for several of these classes. We also consider the related problems pathwidth and interval completion when restricted to some special graph classes.

A set A of vertices of a graph G = (V, E) is an asteroidal set if for each vertex a ɛ A, the set A {a} is contained in one connected component of G−N[a]. The maximum cardinality of an asteroidal set of the graph G is said to be the asteroidal number of G. We show that there are efficient algorithms to compute the asteroidal number for claw-free graphs, HHD-free graphs, circular-arc graphs and circular permutation graphs, while the corresponding decision problem for graphs in general is NP-complete.

On CD-chromatic number and its lower bound in some classes of graphs

Not being (super)thin or solid is hard: A study of grid Hamiltonicity

On relationship between Hamiltonian path and holes in [formula omitted]-coloring of minimum span

A subset T⊆V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every set of vertices of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

A bipartite graph is said to be tree convex if there exists a tree T defined over one part such that the neighborhood of every vertex in the other part forms a connected subtree of T. We study the problem of finding a maximum edge biclique—namely, a complete bipartite subgraph containing the largest possible number of edges—in tree convex bipartite graphs. For the special case of triad convex graphs, where T has exactly three leaves, we present an O(n2 polylog n)-time algorithm, significantly improving the previous best O(n5)-time solution. More generally, for tree convex graphs where the underlying tree has a constant number d of leaves, we present an O(nd-1 polylog n)-time algorithm, which improves the best previously known runtime by a factor of n3 / polylog n for any constant d ≥ 3.

Kombinatorische entscheidungsprobleme: Methoden und anwendungen; fortbildungskurs des instituts für operations research der ETH Zürich: Thomas M. Liebling, Max Rössler (Eds.) Springer, Berlin, 1978, 206 pages, Dfl. 32.75 Soft cover

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

Let $$\mathcal{B}$$B denote the class of all 3-connected cubic bipartite plane graphs. A conjecture of Barnette states that every graph in $$\mathcal{B}$$B has a Hamilton cycle. A cyclic sequence of big faces is a cyclic sequence of different faces, each bounded by at least six edges, such that two faces from the sequence are adjacent if and only if they are consecutive in the sequence. Suppose that $${F_1, F_2, F_3}$$F1,F2,F3 is a proper 3-coloring of the faces of $$G^*\in \mathcal{B}$$G??B. We prove that if every cyclic sequence of big faces of $$G^*$$G? has a face belonging to $$F_1$$F1 and a face belonging to $$F_2$$F2, then $$G^*$$G? has the following properties: $$H^{+-}$$H+-: If any two edges are chosen on the same face, then there is a Hamilton cycle through one and avoiding the other, $$H^{--}$$H--: If any two edges are chosen which are an even distance apart on the same face, then there is a Hamilton cycle that avoids both. Moreover, let $$X$$X and $$Y$$Y partition the set of big faces of $$G^*$$G? such that all such faces of $$F_1$$F1 are in $$X$$X and all such faces of $$F_2$$F2 are in $$Y$$Y. We prove that if every cyclic sequence of big faces has a face belonging to $$X$$X and a face belonging to $$Y$$Y, then $$G^*$$G? has a Hamilton cycle.

A faster FPTAS for knapsack problem with cardinality constraint

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts.In this work we introduce lacon- and shrub-decompositions and use them to characterize transductions of bounded expansion graph classes and other graph classes. If one can efficiently compute sparse shrub- or lacon-decompositions of transductions of bounded expansion classes then one can solve every problem definable in first-order logic in linear time on these classes.

The dominating graph of a graph H has as its vertices all dominating sets of H, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of H. In this paper we prove that the dominating graph of any tree has a Hamilton path. We also show how a result about binary strings leads to a proof that the dominating graph of a cycle on n vertices has a Hamilton path if and only if n is not a multiple of 4.