$K_r$-Saturated Graphs and the Two Families Theorem

A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter, inspired by the well-known Erdos-Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter $mp(G)$. We call $G$ saturated if, for every edge $e$ added to $G$, $mp(G+e) >mp(G)$, and we define $h(n,k)$ to be the least possible number of edges in a saturated graph $G$ on $n$ vertices with $mp(G) < k$, while $mp(G+e) \geq k$ for every new edge $e$. We obtain linear lower and upper bounds for $h(n,k)$, we determine exactly the values of $h(n,k)$ for $k=3$ and $4$, and we present constructions of saturated graphs.

For two given graphs and , a graph is said to be weakly ‐saturated if is a spanning subgraph of which has no copy of as a subgraph and one can add all edges in to in some order so that a new copy of is created at each step. The weak saturation number is the minimum number of edges of a weakly ‐saturated graph. In this paper, we deal with the relation between and , where denotes the Erdős–Rényi random graph and denotes the complete graph on vertices. For any graph and constant , we prove that with high probability. Also, for some graphs , including complete graphs, complete bipartite graphs, and connected graphs with minimum degree or , it is shown that there exists an such that, for every with high probability.

Parameterized approximation algorithms for weighted vertex cover

A pseudoforest is a graph whose connected components have at most one cycle. Let X be a pseudoforest modulator of graph G, i.e. a vertex subset of G such that $$G-X$$ is a pseudoforest. We show that Vertex Cover admits a polynomial kernel being parameterized by the size of the pseudoforest modulator. In other words, we provide a polynomial time algorithm that for an input graph G and integer k, outputs a graph $$G'$$ and integer $$k'$$, such that $$G'$$ has $$\mathcal {O}|X|^{12}$$ vertices and G has a vertex cover of size k if and only if $$G'$$ has vertex cover of size $$k'$$. We complement our findings by proving that there is no polynomial kernel for Vertex Cover parameterized by the size of a modulator to a mock forest a graph where no cycles share a vertex unless $$\text {NP}\subseteq \text {coNP/poly}$$. In particular, this also rules out polynomial kernels when parameterized by the size of a modulator to cactus graphs.

We investigate the following above-guarantee parameterization of the classical Vertex Cover problem: Given a graph $G$ and $k\in\mathbb{N}$ as input, does $G$ have a vertex cover of size at most $(2LP-MM)+k$? Here $MM$ is the size of a maximum matching of $G$, $LP$ is the value of an optimum solution to the relaxed (standard) LP for Vertex Cover on $G$, and $k$ is the parameter. Since $(2LP-MM)\geq{LP}\geq{MM}$, this is a stricter parameterization than those---namely, above-$MM$, and above-$LP$---which have been studied so far. We prove that Vertex Cover is fixed-parameter tractable for this stricter parameter $k$: We derive an algorithm which solves Vertex Cover in time $O^{*}(3^{k})$, pushing the envelope further on the parameterized tractability 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. Our classes are of the form Wk(G), graphs that can be formed by augmenting graphs in G by adding at most k vertices (and incident edges). If G is the class of edgeless graphs, Wk(G) is the class of graphs with a vertex cover of size at most k. We describe a recognition algorithm for Wk(G) running in time O((g + k)|V (G)| + (fk)k), where g and f are modest constants depending on the class G, 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. If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still recognize graphs in Wk(G) in time O(|V (G)|(D + k) + k(D + k)k+3). Our results are obtained by considering minor-closed classes G for which all obstructions are connected graphs, and showing that the size of any obstruction for Wk(G) is O(tk7 + t7k2), where t is a bound on the size of obstructions for G.

Let G=(V, E) be a simple connected graph of order [mid ]V[mid ]=n[ges ]2 and minimum degree δ, and let 2[les ]s[les ]n. We define two parameters, the s-average distance μs(G) and the s-average nearest neighbour distance Λs(G), with respect to each of which V contains an extremal subset X of order s with vertices ‘as spread out as possible’ in G. We compute the exact values of both parameters when G is the cycle Cn, and show how to obtain the corresponding optimal arrangements of X. Sharp upper and lower bounds are then established for Λs(G), as functions of s, n and δ, and the extremal graphs described.

A subset S of vertices of a graph G is k‐dominating if every vertex not in S has at least k neighbors in S. The k‐domination number $\gamma_k(G)$ is the minimum cardinality of a k‐dominating set of G. Different upper bounds on $\gamma_{k}(G)$ are known in terms of the order n and the minimum degree $\delta$ of G. In this self‐contained article, we present an Erdös‐type result, from which some of these bounds follow. In particular, we improve the bound $\gamma_{k}(G) \le (2k- \delta - 1)n/(2k -\delta)$ for $(\delta +3)/2 \le k \le \delta - 1$, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality $\gamma_{k}(G) \le kn/(k +1)$, if $k \le \delta$, of Cockayne et al. This characterization generalizes that of graphs realizing $\gamma_1(G) = \gamma(G) = n/2$. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 33–40, 2008

Let be an integer. Kouider and Lonc proved that the vertex set of every graph G with vertices and minimum degree at least can be covered by cycles. Our main result states that for every and , the same conclusion holds for graphs G with minimum degree that are sparse in the sense that In particular, this allows us to determine the local resilience of random and pseudorandom graphs with respect to having a vertex cover by a fixed number of cycles. The proof uses a version of the absorbing method in sparse expander graphs.

We study approximation algorithms, integrality gaps, and hardness of approximation of two problems related to cycles of “small” length $k$ in a given (undirected) graph. The instance for these problems consists of a graph $G=(V,E)$ and an integer $k$. The $k$-Cycle Transversal problem is to find a minimum edge subset of $E$ that intersects every $k$-cycle. The $k$-Cycle-Free Subgraph problem is to find a maximum edge subset of $E$ without $k$-cycles. Our main result is for the $k$-Cycle-Free Subgraph problem with even values of $k$. For any $k=2r$, we give an $\Omega(n^{-\frac{1}{r}+\frac{1}{r(2r-1)}-\varepsilon})$-approximation scheme with running time $(1/\varepsilon)^{O(1/\varepsilon)}\mathsf{poly}(n)$, where $n=|V|$ is the number of vertices in the graph. This improves upon the ratio $\Omega(n^{-1/r})$ that can be deduced from extremal graph theory. In particular, for $k=4$ the improvement is from $\Omega(n^{-1/2})$ to $\Omega(n^{-1/3-\varepsilon})$. Our additional result is for odd $k$. The $3$-Cycle Transversal problem (covering all triangles) was studied by Krivelevich [Discrete Math., 142 (1995), pp. 281-286], who presented an LP-based $2$-approximation algorithm. We show that $k$-Cycle Transversal admits a $(k-1)$-approximation algorithm, which extends to any odd $k$ the result that Krivelevich proved for $k=3$. Based on this, for odd $k$ we give an algorithm for $k$-Cycle-Free Subgraph with ratio $\frac{k-1}{2k-3}=\frac{1}{2}+\frac{1}{4k-6}$; this improves upon the trivial ratio of $1/2$. For $k=3$, the integrality gap of the underlying LP was posed as an open problem in the work of Krivelevich. We resolve this problem by showing a sequence of graphs with integrality gap approaching $2$. In addition, we show that if $k$-Cycle Transversal admits a $(2-\varepsilon)$-approximation algorithm, then so does the Vertex-Cover problem; thus improving the ratio $2$ is unlikely. Similar results are shown for the problem of covering cycles of length $\leq k$ or finding a maximum subgraph without cycles of length $\leq k$ (i.e., with girth $>k$).

The edge deletion problem (EDP) corresponding to a given class H of graphs is to find the minimum number of edges the deletion of which from a given graph G results in a subgraph G', G' in H. Previous complexity results are extended by showing that the EDP corresponding to any class H of graphs in each of the following cases is NP-hard. (1) H is defined by a set of forbidden homeomorphs or minors in which every member is a 2-connected graph with minimum degree three; (2) BH is defined by K/sub 4/-e as a forbidden homeomorph or minor; and (3) H is defined by P/sub l/, l>or=3, the simple path on l nodes, as a forbidden induced subgraph. >

On the minimum degree forcing F-free graphs to be (nearly) bipartite

Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined as Tr G ′ ( ν ) = ∑ u ∈ V ( G ) 1 d G ( u , ν ) , u ≠ ν . $${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$ The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j), q ij = { 1 + Tr ′ ( ν i ) if i = j and ν i ∈ C Tr ′ ( ν i ) if i = j and ν i ∉ C 1 d ( ν i , ν j ) otherwise $${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} &amp; {{\rm{if}}} &amp; {{\rm{i = j}}} &amp; {{\rm{and}}} &amp; {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} &amp; {{\rm{if}}} &amp; {{\rm{i = j}}} &amp; {{\rm{and}}} &amp; {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} &amp; {{\rm{otherwise}}} &amp; {} &amp; {} &amp; {} \cr } } \right.$$ where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.

Corradi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all \(k\ge 1\) and \(n\ge 3k\), every (simple) graph G on n vertices with minimum degree \(\delta (G)\ge 2k\) contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corradi–Hajnal theorem: For all \(k\ge 1\) and \(n\ge 3k\), every graph G on n vertices contains k disjoint cycles, provided that \(d(x)+d(y)\ge 4k-1\) for all distinct nonadjacent vertices x, y. Very recently, it was refined for \(k\ge 3\) and \(n\ge 3k+1\): If G is a graph on n vertices such that \(d(x)+d(y)\ge 4k-3\) for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number \(\alpha (G)\le n-2k\) and G is not one of two small exceptions in the case \(k=3\). But the most difficult case, \(n=3k\), was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corradi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.