Edgeless Graphs Research Articles

Essentially Tight Kernels for (Weakly) Closed Graphs

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number gamma (Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to the standard parameter solution size k. The weak closure number gamma of a graph is upper-bounded by the minimum of its closure number c and its degeneracy d. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size k^{mathcal {O}(gamma )}, k^{mathcal {O}(gamma )}, and (gamma k)^{mathcal {O}(gamma )}, respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size k^{o(gamma )} by previous results on their kernelization complexity on degenerate graphs (Cygan et al. ACM Trans Algorithms 13(3):43:1–43:22, 2017). For Capacitated Vertex Cover, we show that even a kernel of size k^{o(c)} is unlikely. In contrast, for Connected Vertex Cover, we obtain a kernel with mathcal {O}(ck^2) vertices. Moreover, we prove that searching for an induced subgraph of order at least k belonging to a hereditary graph class mathcal {G} admits a kernel of size k^{mathcal {O}(gamma )} when mathcal {G} contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number c and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class Π. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most k from a graph with maximum degree 1 can be computed in polynomial time when k is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree 2 or connected components of size at most c, c ≥ 3, is NP-hard. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably has no f(k) · |I|O(1)-time algorithm and that, in contrast, an optimal network with at most k arcs can be computed in 2O(k) · |I|O(1) time where |I| is the total input size.

Heuristic method to determine lucky k-polynomials for k-colorable graphs

Abstract The existence of edges is a huge challenge with regards to determining lucky k-polynomials of simple connected graphs in general. In this paper the lucky 3-polynomials of path and cycle graphs of order, 3 ≤ n ≤ 8 are presented as the basis for the heuristic method to determine the lucky k-polynomials for k-colorable graphs. The difficulty of adjacency with graphs is illustrated through these elementary graph structures. The results are also illustratively compared with the results for null graphs (edgeless graphs). The paper could serve as a basis for finding recurrence results through innovative methodology.

Complexity dichotomies for the Minimum[formula omitted]-Overlay problem

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlaysF on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph. While it is easy to see that the complete graph on |V(H)| overlays F on a hypergraph H whenever the problem admits a solution, the MinimumF-Overlay problem asks for such a graph with at most k edges, for some given k∈N. This problem allows to generalize some natural problems which may arise in practice. For instance, if the family F contains all connected graphs, then MinimumF-Overlay corresponds to the Minimum Connectivity Inference problem (also known as Subset Interconnection Design problem) introduced for the low-resolution reconstruction of macro-molecular assembly in structural biology, or for the design of networks.Our main contribution is a strong dichotomy result regarding the polynomial vs. NP-complete status with respect to the considered family F. Roughly speaking, we show that the easy cases one can think of (e.g. when edgeless graphs of the right sizes are in F, or if F contains only cliques) are the only families giving rise to a polynomial problem: all others are NP-complete. We then investigate the parameterized complexity of the problem and give similar sufficient conditions on F that give rise to W[1]-hard, W[2]-hard or FPT problems when the parameter is the size of the solution. This yields an FPT/W[1]-hard dichotomy for a relaxed problem, where every hyperedge of H must contain some member of F as a (non necessarily spanning) subgraph.

Tight Running Time Lower Bounds for Vertex Deletion Problems

For a graph class Π, the Π-V ertex D eletion problem has as input an undirected graph G = ( V , E ) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-V ertex D eletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-V ertex D eletion does not admit a 2 o ( n ) -time algorithm where n is the number of vertices in G . We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2 o ( n+m ) -time algorithm for Π-V ertex D eletion . On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2 O ( n ) time or 2 o ( m ) time, then Π-V ertex D eletion can be solved in 2 O (√ m ) + O ( n ) or 2 o ( m ) + O ( n ) time, respectively. We also consider restrictions on the domain of the input graph G . For example, we obtain that Π-V ertex D eletion cannot be solved in 2 o (√ n ) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.

Auxiliary embeddings and constructing triangular embeddings of joins of complete graphs with edgeless graphs

Until recently there were no results on the orientable or nonorientable genus of graphs Kn+Km¯ for n/2≤m<n−1. Recently, using index one current graphs, McCourt (2014) constructed a nonorientable triangular embedding of K36t+3+K18t+1+6i¯ for i=0,1,…,t−1 and t≥1. In the present paper, using index three current graphs, we construct auxiliary embeddings that allow three arbitrary triangular embeddings of Kn+1 to be incorporated into a triangular embedding of K3n+Km¯ for many cases where m can lie anywhere in the interval 1≤m≤2n. Using these auxiliary embeddings we construct the following six classes of triangular embeddings: orientable triangular embeddings of K36t+6+K1+2i¯ for t≥1 and i=0,1,…,12t, K36t+18+K1+2i¯ for t≥0 and i=0,1,…,12t+4, and nonorientable triangular embeddings of K18t+K1+2i¯ for t≥1 and i=0,1,…,6t−2, K18t+6+K1+2i¯ for t≥1 and i=0,1,…,6t, K18t−3+Ki¯ for t≥2 and i=3,4,…,12t−6, K18t+9+Ki¯ for t≥1 and i=3,4,…,12t+2. Using these auxiliary embeddings we also show that for a certain positive constant a and for an infinite number of values of t, the number of nonisomorphic orientable triangular embeddings of K36t+6+K1+2i¯ for t≥1 and every i∈{0,1,…,12t} is at least (36t+6)a(36t+6)2−o((36t+6)2) as t→∞. Analogous results are obtained for the other five classes of triangular embeddings as well.

Graph Divisible Designs and Packing Constructions

We introduce a generalization of group divisible designs and offer example applications to challenging problems in design theory. The generalization considers edge-decompositions of joins of arbitrary graphs, whereas group divisible designs handle only joins of edgeless graphs. Our example constructions include: (1) optimal packings with block size five for the previously unsettled congruence class $$v \equiv 13 \pmod {20}$$v?13(mod20); (2) an optimal grooming with with ratio seven for the previously unsettled congruence class $$v \equiv 56 \pmod {84}$$v?56(mod84); and (3) a constructive `quadratic' embedding of partial designs with block size four.

Edgeless graphs are the only universal fixers

Given two disjoint copies of a graph G, denoted G1 and G2, and a permutation π of V (G), the graph πG is constructed by joining u ∈ V (G1) to π(u) ∈ V (G2) for all u ∈ V (G1). G is said to be a universal fixer if the domination number of πG is equal to the domination number of G for all π of V (G). In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely.

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph K m ¯ + K n with m ≥ n − 1 , the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph K m ¯ + K n ¯ = K m , n . The orientable genus problem for K m ¯ + K n with m ≥ n − 1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of K m ¯ + K n when n is even and m ≥ n , the genus of K m ¯ + K n when n = 2 p + 2 for p ≥ 3 and m ≥ n − 1 , and the genus of K m ¯ + K n when n = 2 p + 1 for p ≥ 3 and m ≥ n + 1 . In all of these cases the genus is the same as the genus of K m , n , namely ⌈ ( m − 2 ) ( n − 2 ) / 4 ⌉ .

Claw-free graphs are not universal fixers

For any permutation π of the vertex set of a graph G , the generalized prism π G is obtained by joining two copies of G by the matching { u π ( u ) : u ∈ V ( G ) } . Denote the domination number of G by γ ( G ) . If γ ( π G ) = γ ( G ) for all π , then G is called a universal fixer. The edgeless graphs are the only known universal fixers, and are conjectured to be the only universal fixers. We prove that claw-free graphs are not universal fixers.

On Equality in an Upper Bound for the Domination Number with Respect to Nondegenerate Properties

For a graph property P and a graph G, a subset S of vertices of G is a P-set if the subgraph induced by S has the property P. The domination number with respect to the property P, denoted by γP (G), is the minimum cardinality of a dominating P-set. Any property P satisfied by all edgeless graphs is called nondegenerate. For any graph G with n vertices and maximum degree Δ(G), γP (G) ≤ n - Δ(G) where P is nondegenerate and closed under union with K1 In this paper we characterize the connected graphs and the connected triangle-free graphs which achieve this upper bound.

The nonorientable genus of joins of complete graphs with large edgeless graphs

We show that for n = 4 and n ⩾ 6 , K n has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, K m ¯ + K n = K m + n − K m , and show that for n ⩾ 3 and m ⩾ n − 1 its nonorientable genus is ⌈ ( m − 2 ) ( n − 2 ) / 2 ⌉ except when ( m , n ) = ( 4 , 5 ) . We then extend these results to find the nonorientable genus of all graphs K m ¯ + G where m ⩾ | V ( G ) | − 1 . We provide a result that applies in some cases with smaller m when G is disconnected.

Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class W k ( G ) , where for each graph G in W k ( G ) , the removal of a set of at most k vertices from G results in a graph in the base graph class G . (If G is the class of edgeless graphs, W k ( G ) is the class of graphs with bounded vertex cover.) When G is a minor-closed class such that each graph in G has bounded maximum degree, and all obstructions of G (minor-minimal graphs outside G ) are connected, we obtain an O ( ( g + k ) | V ( G ) | + ( fk ) k ) recognition algorithm for W k ( G ) , where g and f are constants (modest and quantified) depending on the class G . If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still obtain a running time of O ( | V ( G ) | ( D + k ) + k ( D + k ) k + 3 ) for recognition of graphs in W k ( G ) . Our results are obtained by considering bounded-degree minor-closed classes for which all obstructions are connected graphs, and showing that the size of any obstruction for W k ( G ) is O ( tk 7 + t 7 k 2 ) , where t is a bound on the size of obstructions for G . A trivial corollary of this result is an upper bound of ( k + 1 ) ( k + 2 ) on the number of vertices in any obstruction of the class of graphs with vertex cover of size at most k. These results are of independent graph-theoretic interest.

Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

New Results on Generalized Graph Coloring

For graph classes \wp_1,...,\wp_k, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V_1,...,V_k so that V_j induces a graph in the class \wp_j (j=1,2,...,k). If \wp_1=...=\wp_k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k≥ 3. Recently, this result has been generalized by showing that if all \wp_i's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the \wp_i's are co-additive.

(n,e)-Graphs with maximum sum of squares of degrees*

Among all simple graphs on n vertices and e edges, which ones have the largest sum of squares of the vertex degrees? It is easy to see that they must be threshold graphs, but not every threshold graph is optimal in this sense. Boesch et al. [Boesch et al., Tech Rep, Stevens Inst Tech, Hoboken NJ, 1990] showed that for given n and e there exists exactly one graph of the form G1(p,q,r) = Kp + (Sq ∪ K1,r) and exactly one G2(p,q,r) = and that one of them is optimal, where K and S indicate complete and edgeless graphs, K1,r indicates a star on r + 1 vertices, ∪ indicates disjoint union, and + indicates complete disjoint join. We specify a general threshold graph in the form G(a,b,c,d, …) = Ka + (Sb ∪ (Kc + (Sd ∪ ···))) or its complement G(a,b,c,d, …), and we prove that every optimal graph has the form G(a,b,c,d) or G(a,b,c,d) with b ⩽ 1 or c ⩽ 1 or d ⩽ 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 283–295, 1999

(n,e)‐Graphs with maximum sum of squares of degrees*

Among all simple graphs on n vertices and e edges, which ones have the largest sum of squares of the vertex degrees? It is easy to see that they must be threshold graphs, but not every threshold graph is optimal in this sense. Boesch et al. [Boesch et al., Tech Rep, Stevens Inst Tech, Hoboken NJ, 1990] showed that for given n and e there exists exactly one graph of the form G1(p,q,r) = Kp + (Sq ∪ K1,r) and exactly one G2(p,q,r) = and that one of them is optimal, where K and S indicate complete and edgeless graphs, K1,r indicates a star on r + 1 vertices, ∪ indicates disjoint union, and + indicates complete disjoint join. We specify a general threshold graph in the form G(a,b,c,d, …) = Ka + (Sb ∪ (Kc + (Sd ∪ ···))) or its complement G(a,b,c,d, …), and we prove that every optimal graph has the form G(a,b,c,d) or G(a,b,c,d) with b ⩽ 1 or c ⩽ 1 or d ⩽ 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 283–295, 1999

Generalized chromatic numbers of random graphs

AbstractLet p be a hereditary graph property. The p‐chromatic number of a graph is the minimal number of classes in a vertex partition wherein each class spans a subgraph with property p. For the property p of edgeless graphs the p‐chromatic number is just the usual chromatic number, whose value is known to be (1/2 + o(1))n/log2 n for almost every graph of order n. We show that we may associate with every nontrivial hereditary property p an explicitly defined natural number r = r(p), and that the p‐chromatic number is then (l/2r + o(1))n/log2 n for almost every graph of order n. © 1995 John Wiley &amp; Sons, Inc.

Centroids and medians of finite metric spaces

AbstractThe median of a weighted finite metric space consists of the points minimizing the total weighted distance to the points of the space. The centroid is formed by the points p satisfying the following minimax condition: the maximal weight of a geodesically convex set not containing a point X attains its minimum at p. It is well known that in a tree network the centroid and the median coincide for every distribution of weights. The metric spaces for which the latter property is characteristic are determined in this paper. These spaces are obtained from three classess of graphs: median graphs, joins of complete graphs with edgeless graphs, and joins of two‐vertex edgeless graphs.