Let $G = (V, E)$ be a simple graph, and let $f: V \to \{0, 1, 2, 3\}$ be a function. A vertex \( u \) is considered an undefended vertex with respect to \( f \) if \( f(u) = 0 \) and there is no adjacent vertex \( v \) satisfying \( f(v) \geq 2 \). A function \( f \) is termed a generous Roman dominating function (GRD-function) if, for every vertex \( u \) with \( f(u) = 0 \), there exists at least one adjacent vertex \( v \) such that \( f(v) \geq 2 \) and the modified function \( f': V \to \{0,1,2,3\} \), defined as\( f'(u) = \alpha, \quad f'(v) = f(v) - \alpha,\) where \( \alpha \in \{1,2\} \), and\(f'(w) = f(w) \quad \text{for all } w \in V \setminus \{u, v\},\) ensures that no vertex remains undefended. The weight of a GRD-function \( f \) is defined as\(f(V) = \sum_{u \in V} f(u).\) The smallest possible weight of a GRD-function on \( G \) is known as the generous Roman domination number of \( G \), denoted by \( \gamma_{gR}(G) \). The generous Roman domination subdivision number, represented as \( \mathrm{sd}_{\gamma_{gR}}(G) \), is the minimum number of edges that must be subdivided (where each edge in \( G \) can be subdivided at most once) to increase the generous Roman domination number. In this paper, we establish upper bounds on the generous Roman domination subdivision number. Furthermore, we determine the exact value of this parameter for certain families of graphs, including paths, cycles, and ladders. Further, we present several sufficient conditions for a graph \( G \) to have a small value of \( sd_{\gamma_{gR}}(G) \).\end{abstract}\keywords{generous Roman domination, generous Roman domination subdivision number}

The paper extends the concept of bondage numbers to certified domination, introducing the certified bondage number of a graph. A certified dominating set R is a dominating set of a graph H, if every vertex in R has either zero or at least two neighbours in V\R, where V is the vertex set of H. The minimum cardinality of certified dominating set of H is the certified domination number of H denoted by γcer(H). The bondage number b(H) is defined to be the cardinality of least number of edges F ⊂ E(H) such that γ(H − F) > γ(H). Motivated by this parameter, we extended this concept on certified domination number and defined certified bondage number of a graph H, b+cer(H) [b−cer(H)] to be the cardinality of the least number of edges F ⊂ E(H) such that γcer(H−F) > γcer(H) [γcer(H−F) < γcer(H)] that is minimum number of edges to be removed to increase (or decrease) the certified domination number of H. In this paper, we establish the values of certified bondage number for generalised Petersen graphs P(n, k), where k = 1, 2, as well as for certain classes of graphs.

A certified dominating set S is a dominating set of a graph G, if every vertex in S has either zero or at least two neighbours in V \S. The minimum cardinality of certified dominating set of G is the certified domination number of G denoted by γcer(G). We defined certified domination subdivision number Sd+γcer (G) [Sd−γcer (G)] of a graph G to be the minimum number of edges thatmust be subdivided (where no edge in G can be subdivided more than once) in order to construct a graph with a certified domination number larger [lesser] than the certified domination number of G. In this paper, we determine the values of certified domination subdivision number for certain classes of graphs including circulant graphs [Cn(1, 2) and Cn(1, 3)] and petersen graphs [P(n, 1) and P(n, 2)].

Abstract Given a graph F and a positive integer n , the weak F -saturation number $${\textrm{wsat}}(K_n,F)$$ wsat ( K n , F ) is the minimum number of edges in a graph H on n vertices such that the edges missing in H can be added, one at a time, so that every edge creates a copy of F . Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. Let W be a vector space spanned by vectors w ( e ) assigned to edges e of $$K_n$$ K n . Suppose that, for every copy $$F'\subset K_n$$ F ′ ⊂ K n of F , there exist non-zero scalars $$\lambda _e$$ λ e , $$e\in E(F')$$ e ∈ E ( F ′ ) , satisfying $$\sum _{e\in E(F')}\lambda _e w(e)=0$$ ∑ e ∈ E ( F ′ ) λ e w ( e ) = 0 . Then $$\textrm{dim}W\le {\textrm{wsat}}(K_n,F)$$ dim W ≤ wsat ( K n , F ) . In this paper, we prove limitations of this approach: we find infinitely many F such that, for every vector space W as above, $$\textrm{dim}W<{\textrm{wsat}}(K_n,F)$$ dim W < wsat ( K n , F ) . We also introduce a modification of this approach that yields tight lower bounds even when the original direct approach is insufficient. Finally, we generalise our results to random graphs, complete multipartite graphs, and hypergraphs.

ABSTRACTA well‐known theorem of Mantel states that every ‐vertex graph with more than edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erdős, Faudree, and Rousseau (1992) showed that a graph on vertices with more than edges contains at least edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erdős, Faudree, and Rousseau. Using the supersaturation‐stability and the spectral technique, we prove that every ‐vertex graph with contains at least triangular edges, unless is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation‐stability can be used to revisit a conjecture of Erdős concerning the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadžiivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order of the spectral extremal graph when we forbid the friendship graph as a substructure. We drop the condition that requires the order to be sufficiently large, which was investigated by Cioabă et al. (2020) using the triangle removal lemma. Thirdly, this method can be utilized to deduce the classical stability for odd cycles, and it gives more concise bounds on parameters. Finally, supersaturation stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).

The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph $G$. Our goal is to find a spanning subgraph $S$ of $G$ with the minimum number of edges which is $2$-vertex-connected, namely $S$ remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is $10/7$ by Heeger and Vygen [SIDMA'17] (improving on earlier results by Khuller and Vishkin [STOC'92] and Garg, Vempala and Singla [SODA'93]). Here we present an improved $4/3$ approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are ``almost'' 3-vertex-connected. The latter reduction might be helpful in future work.Comment: 44 pages. This is the TheoretiCS journal version

ABSTRACTFor graphs and , the saturation number is the minimum number of edges in an inclusion‐maximal ‐free subgraph of . In 2017, Korándi and Sudakov initiated the study of saturation in random graphs. They showed that for constant , whp . We show that for every graph and every constant , whp . Furthermore, if every edge of belongs to a triangle, then the above is the right asymptotic order of magnitude, that is, whp . We further show that for a large family of graphs with an edge that does not belong to a triangle, which includes all bipartite graphs, for every and constant , whp . We conjecture that this sharp transition from to depends only on this property, that is, that for any graph with at least one edge that does not belong to a triangle, whp . We further generalize the result of Korándi and Sudakov, and show that for a more general family of graphs , including all complete graphs and all complete multipartite graphs of the form , for every and every constant , whp . Finally, we show that for every complete multipartite graph and every , .

In this paper, we initiate the study of quantum algorithms in the Graph Drawing research area. We focus on two foundational drawing standards: 2-level drawings and book layouts. Concerning $2$-level drawings, we consider the problems of obtaining drawings with the minimum number of crossings, $k$-planar drawings, quasi-planar drawings, and the problem of removing the minimum number of edges to obtain a $2$-level planar graph. Concerning book layouts, we consider the problems of obtaining $1$-page book layouts with the minimum number of crossings, book embeddings with the minimum number of pages, and the problem of removing the minimum number of edges to obtain an outerplanar graph. We explore both the quantum circuit and the quantum annealing models of computation. In the quantum circuit model, we provide an algorithmic framework based on Grover's quantum search, which allows us to obtain, ignoring polynomial terms, a quadratic speedup on the best known classical exact algorithms for all the considered problems. In the quantum annealing model, we perform experiments on the quantum processing unit provided by D-Wave, focusing on the classical $2$-level crossing minimization problem, demonstrating that quantum annealing is competitive with respect to classical algorithms.

Abstract Analysis of the reliability of networks is crucial to the design and optimization of networks. The $P$-conditional edge-connectivity $\lambda (P;G)$ of a network $G$ is the minimum number of edges whose deletion will divide $G$ into several components, and each component satisfies the property $P$. It proposes a more pinpoint analysis to the reliability of networks. The paper investigates several different the $P$-conditional edge-connectivities of $K_{p;t}^{n}$, such as $d$-embedded edge-connectivity $\lambda (P_{1}^{d};K_{p;t}^{n})$, $(t-1)pd$-good-neighbor edge-connectivity $\lambda (P_{3}^{(t-1)pd};K_{p;t}^{n})$, $(t-1)pd$-average edge-connectivity $\lambda (P_{4}^{(t-1)pd};K_{p;t}^{n})$ for $t\geq 2$, $p\geq 2$ and $0\leq d\leq n-1$, and they possess the same value $(t-1)p(n-d)(tp)^{d}$. Besides, we derive $\lambda (P_{2}^{l}; K_{p;t}^{n})= (t-1)pnl-ex_{l}(K_{p;t}^{n})$ for $1\leq l\leq (tp)^{{\lfloor \frac{n}{2} \rfloor }}$, where $ex_{l}(K_{p;t}^{n})$ denotes the twice of the maximum number of edges in a subgraph induced by $l$ vertices in $K_{p;t}^{n}$. Our method generalizes the result of Yu and Xu in (Comput J 2024; 67: 688–93) and Zhang et al. in (J Supercomput 2022; 78: 7936–47).

Let [Formula: see text] and [Formula: see text] be graphs, where we view [Formula: see text] as the “host” graph and [Formula: see text] as a “forbidden” graph. A spanning subgraph [Formula: see text] of [Formula: see text] is called [Formula: see text]-saturated in[Formula: see text] if [Formula: see text] contains no subgraph isomorphic to [Formula: see text], but [Formula: see text] contains [Formula: see text] for any edge [Formula: see text]. We let [Formula: see text] be the minimum number of edges in any graph [Formula: see text] which is [Formula: see text]-saturated in [Formula: see text], where [Formula: see text] if [Formula: see text] contains no copy of [Formula: see text] as a subgraph. Let [Formula: see text] be the [Formula: see text]-dimensional mesh (or grid), with vertex set [Formula: see text] integer, [Formula: see text] and edge set [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the star graph on [Formula: see text] leaves. In this paper we study [Formula: see text]. We give asymptotically exact results for [Formula: see text]. We also give upper bounds for [Formula: see text], [Formula: see text], which are within a factor of [Formula: see text] from optimal when [Formula: see text]. These two results are based on constructions, as well as lower bounds we obtain for [Formula: see text] for [Formula: see text] and [Formula: see text]. Finally for arbitrary [Formula: see text] we obtain asymptotically exact results for [Formula: see text], thereby showing the asymptotic behavior of the minimum size of a maximal matching in [Formula: see text]. This result is based on a construction for the upper bound, and an edge weighting argument for the matching lower bound.

ABSTRACTLet denote the minimum number of edges whose addition to results in a Hamiltonian graph, and let denote the minimum number of edges whose addition to results in a pancyclic graph. We study the distributions of in the context of binomial random graphs. Letting , we prove that there exists a function of order such that, if with , then with high probability . Let denote the number of degree vertices in . A trivial lower bound on is given by the expression . We show that in the random graph process with high probability there exist times , both of order , such that for every and for every . The time can be characterized as the smallest for which contains less than copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of with high probability. For the binomial random graph, this implies that if and , then, with high probability, . For completion to pancyclicity, we show that if and , then, with high probability, . Finally, we present a polynomial time algorithm such that, if and , then, with high probability, the algorithm returns a set of edges of size whose addition to results in a pancyclic (and therefore also Hamiltonian) graph.

$$(K_{1}\vee {P_{t})}$$-saturated graphs with minimum number of edges

ABSTRACTFor a fixed graph , we say that an edge‐colored graph is weakly ‐rainbow saturated if there exists an ordering of such that, for any list of pairwise distinct colors from , the nonedges in color can be added to , one at a time, so that every added edge creates a new rainbow copy of . The weak rainbow saturation number of , denoted by , is the minimum number of edges in a weakly ‐rainbow saturated graph on vertices. In this paper, we show that for any nonempty graph , the limit exists. This answers a question of Behague et al. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if contains no pendant edges.

A vertex subset S of a graph G is an independent set if no two vertices in S are adjacent and is a dominating set if every vertex not in S is adjacent to a vertex in S. If S is both independent and dominating in G, then S is said to be an independent dominating set. The independent domination number of G is the minimum cardinality among all independent dominating sets of G. In this paper, we investigate the independent bondage number of G defined as the minimum number of edges whose removal from G results in a graph with a greater independent domination number. We prove that the independent bondage number is at most 5 (respectively, 6, 7) for planar graphs with minimum degree at least 3 without cycles of lengths 4 and 5 (respectively, without cycles of length 4, without intersecting triangles). All these results improve two earlier bounds for planar graphs.

A graph \(G\) whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. There exist infinite families of graphs that are not TI-graphs. We define the TI-augmentation number \(\operatorname{ti}(G)\) of a graph \(G\) to be the minimum number of edges that must be added to \(G\) to ensure that the resulting graph is a TI-graph. We show that every tree \(T\) of order \(n \geq 5\) satisfies \(\operatorname{ti}(T) \leq \frac{1}{5}n\). We prove that if \(G\) is a bipartite graph of order \(n\) with minimum degree \(\delta(G) \geq 3\), then \(\operatorname{ti}(G) \leq \frac{1}{4}n\), and if \(G\) is a cubic graph of order \(n\), then \(\operatorname{ti}(G) \leq \frac{1}{3}n\). We conjecture that \(\operatorname{ti}(G) \leq \frac{1}{6}n\) for all graphs \(G\) of order \(n\) with \(\delta(G) \geq 3\), and show that there exist connected graphs \(G\) of sufficiently large order \(n\) with \(\delta(G) \geq 3\) such that \(\operatorname{ti}(T) \geq (\frac{1}{6} - \varepsilon) n\) for any given \(\varepsilon \gt 0\).

Let \(\mathcal{F}\) be a family of graphs, and \(H\) a ``host'' graph. A spanning subgraph \(G\) of \(H\) is called \(\mathcal{F}\)- saturated in \(H\) if \(G\) contains no member of \(\mathcal{F}\) as a subgraph, but \(G+e\) contains a member of \(\mathcal{F}\) for any edge \(e\in E(H) - E(G)\). We let \(Sat(H,\mathcal{F})\) be the minimum number of edges in any graph \(G\) which is \(\mathcal{F}\)-saturated in \(H\), where \(Sat(H,\mathcal{F}) = |E(H)|\) if \(H\) contains no member of \(\mathcal{F}\) as a subgraph. Let \(P_{m}^{r}\) be the \(r\)-dimensional grid, with entries in each coordinate taken from \(\{1,2,\cdots , m\}\), and \(K_{t}\) the complete graph on \(t\) vertices. Also let \(S(F)\) be the family of all subdivisions of a graph \(F\). There has been substantial previous work on extremal questions involving subdivisions of graphs, involving both \(Sat(K_{n},S(F))\) and the Turan function \(ex(K_{n},S(F))\), for \(F = K_{t}\) or \(F\) a complete bipartite graph. In this paper we study \(Sat(H, S(F))\) for the host graph \(H = P_{m}^{r}\), and \(F = K_{4}\), motivated by previous work on \(Sat(K_{n}, S(K_{t}))\). Our main results are the following; 1) If at least one of \(m\) or \(n\) is odd with \(m\geq 5\) and \(n\geq 5\), then \(Sat(P_{m}\times P_{n}, S(K_{4})) = mn + 1.\) 2) For \(m\) even and \(m\geq 4\), we have \(m^{3} + 1 \le Sat(P_{m}^{3}, S(K_{4}))\le m^{3} + 2.\) 3) For \( r\geq 3\) with \(m\) even and \(m\geq 4\), we have \(Sat(P_{m}^{r}, S(K_{4})) \le m^{r} + 2^{r-1} - 2\).

By the skewness of a graph, we mean the minimum number of its edges whose deletion results in a planar graph. We determine the skewness of a large family of cubic bipartite graphs (which includes the Heawood graph as a special case). Moreover, we also determine those classes of these cubic graphs which are π -skew whose resulting plane graphs (upon deleting the right minimum number of edges) are hexagulations.

We say that a graph $G$ is $K_r$-saturated if $G$ contains no copy of $K_r$ but adding any new edge to $G$ creates a copy of $K_r$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices with minimum degree at least $t$. Day showed that for fixed $r \geq 3$ and $t \geq r-2$, $sat(n,K_r,t)=tn-c(r,t)$ for large enough $n$, where $c(r,t)$ is a constant depending on $r$ and $t$, and proved the bounds $$ 2^t t^{3/2} \ll_r c(r,t) \leq t^{t^{2t^2}} $$for fixed $r$ and large $t$. In this paper we show that for fixed $r$ and large $t$, the order of magnitude of $c(r,t)$ is given by $c(r,t)=\Theta_r \left(4^t t^{-1/2} \right)$. Moreover, we investigate the dependence on $r$, obtaining the estimates$$ \frac{4^{t-r}}{\sqrt{t-r+3}} + r^2 \ll c(r,t) \ll \frac{4^{t-r} \min{(r,\sqrt{t-r+3})}}{\sqrt{t-r+3}} + r^2 \ . $$We further show that for all $r$ and $t$, there is a finite collection of graphs such that all extremal graphs are blow-ups of graphs in the collection. Using similar ideas, we show that every large $K_r$-saturated graph with $e$ edges has a vertex cover of size $O(e / \log e)$, uniformly in $r \geq 3$. This strengthens a previous result of Pikhurko. We also provide examples for which this bound is tight. A key ingredient in the proofs is a new version of Bollobás's Two Families Theorem.

In a rainbow version of the classical Turán problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which guarantees existence of a rainbow copy (having at most one edge from each graph) of a given graph. Here, we prove an optimal solution for this problem for any directed star and any number of colors.

The question on how to colour a graph G when the number of available colours to colour G is less than that of the chromatic number χ ( G ) , such that the resulting colouring gives a minimum number of edges whose end vertices have the same colour, has been a study of great interest. Such an edge whose end vertices receive the same colour is called a bad edge. In this paper, we use the concept of δ ( k ) -colouring, where 1 ≤ k ≤ χ ( G ) − 1 , which is a near proper colouring that permits a single colour class to have adjacency between the vertices in it and restricts every other colour class to be an independent set, to find the minimum number of bad edges obtained from the same for some wheel-related graphs. The minimum number of bad edges obtained from δ ( k ) -colouring of any graph G is denoted by b k ( G ) .