Graphs

Klavík et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals \(\mathfrak{L}_v\) and \(\mathfrak{R}_v\) called bounds. We ask whether there exists a bounded representation in which each interval \(\mathfrak{I}_v\) has its left endpoint in \(\mathfrak{L}_v\) and its right endpoint in \(\mathfrak{R}_v\). We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs.Robert’s Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly, the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klavík et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently.The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

Let G be a connected simple graph on n vertices. The Gallai's conjecture asserts that the edges of G can be decomposed into paths. In this paper, we show that the conjecture is true if G is a maximal outerplanar graph or G ia a 2-connected outerplanar graph.

Thin strip graphs

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring \(f:V(G) \rightarrow \{r,b\}\) of G such that for every \(i \in \{r,b\}\), there are at least k edges with both end vertices colored i. It is known that the problem is \(\mathsf {NP} \)-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that 2-Load Coloring cannot be solved in time \(f(w)n^{o(w)}\), unless ETH fails and it can be solved in time \(n^{O(w)}\), where n is the size of the input graph, w is the clique-width of the graph and f is an arbitrary function of w. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that 2-Load Coloring is \(\mathsf {NP} \)-complete even on bipartite graphs and split graphs.

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

This paper addresses the classic online Steiner tree problem on edge-weighted graphs. It is known that a greedy (nearest neighbor) online algorithm has a tight competitive ratio for wide classes of graphs, such as trees, rings, any class including series-parallel graphs, and unweighted graphs with bounded diameter. However, we do not know any greedy or non-greedy tight deterministic algorithm for other classes of graphs. In this paper, we observe that a greedy algorithm is $$\Omega (\log n)$$ -competitive on outerplanar graphs, where n is the number of vertices, and propose a 5.828-competitive deterministic algorithm on outerplanar graphs. Our algorithm connects a requested vertex and the tree constructed thus far using a path that is constant times longer than the distance between them. We also present a lower bound of 4 for arbitrary deterministic online Steiner tree algorithms on outerplanar graphs.

The flooding games, which are called Flood-It, Mad Virus, or HoneyBee, are a kind of coloring games and they have been becoming popular online. In these games, each player colors one specified cell in his/her turn, and all connected neighbor cells of the same color are also colored by the color. This flooding or coloring spreads on the same color cells. It is natural to consider these new coloring games on more general boards, or general graphs. Recently, computational complexities of the variants of the flooding games on several graph classes have been studied. In this paper, we investigate the flooding games on some graph classes characterized by interval representations. Our results state that the number of colors is a key parameter to determine the computational complexity of the flooding games. When the number of colors is a fixed constant, these games can be solved in polynomial time on an interval graph. On the other hand, if the number of colors is not bounded, the flooding game is NP-complete on a proper interval graph. We also state similar results for split graphs.

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

Max point-tolerance graphs

Computing a minimum outer-connected dominating set for the class of chordal graphs

A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let ar(G,H) denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by kK2 a matching of size k and On the class of all maximal outer-planar graphs on n vertices, respectively. The outer-planar anti-Ramsey number of graph H, denoted by ar(On,H), is defined as max{ar(On,H)|On∈On}. It seems nontrivial to determine the exact values for ar(On,H) because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that ar(On,kK2)≤n+3k−8 for all n≥2k and k≥6, which improves the existing upper bound for ar(On,kK2), and prove that ar(On,kK2)=n+2k−5 for n=2k and k≥5. We also obtain that ar(On,6K2)=n+6 for all n≥29.

Polar graphs and maximal independent sets

A special class of graphs is introduced in this paper. The graphs belonging to this class are characterised by the existence of unique node labels. A number of matching algorithms for graphs with unique node labels are developed. It is shown that problems such as graph isomorphism, subgraph isomorphism, maximum common subgraph (MCS) and graph edit distance (GED) have a computational complexity that is only quadratic in the number of nodes. Moreover, computing the median of a set of graphs is only linear in the cardinality of the set. In a series of experiments, it is demonstrated that the proposed algorithms run very fast in practice. The considered class makes the matching of large graphs, consisting of thousands of nodes, computationally tractable. We also discuss an application of the considered class of graphs and related matching algorithms to the classification and detection of abnormal events in computer networks.