For a graph G, we call a set D⊆V(G) a total isolating set of G if G[D] has no isolated vertices and G−N[D] contains no edges, where G[D] is the induced subgraph of G on D and N[D] is the closed neighborhood of D. The total isolation number ιt(G) of G is the minimum cardinality of a total isolating set in G. Recently, Boyer, Goddard and Henning proved that if G is a connected graph with n ≥ 4 vertices and G¬≅C7, then ιt(G)≤n2 and the bound is tight. In this paper, we classify all extremal graphs which achieve the bound. We show that excluding an infinite family of graphs, there exist exactly thirteen sporadic extremal graphs. This solves an open problem proposed by Boyer, Goddard and Henning.

Let G be a graph and .F be a family of graphs. A subset D of vertices of G is called an .F-isolating set of G if G-N[D] contains no member of .F as a subgraph, where N[D] is the closed neighborhood of D. The .F-isolation number of G, denoted by t(G, .F), is the minimum cardinality of an .F-isolating set of G. For any integer k > 3, let C-k be the cycle of length k and C-k be the set of cycles of length at least k. In 2020, Borg (2020) [2] proved that if G is a connected n-vertex graph that is not a triangle, then t(G, {C-3}) <= t(G, C-3) <= n/4 . Very recently, Bartolo, Borg and Scicluna (2024) [1] further proved that if G is a connected n-vertex graph that is not one of the nine exceptional graphs, then t(G, {C4}) <= n5 . In the same paper, they also conjectured that if G is a connected n-vertex graph, then t(G, C4) <= n/5 unless G belongs to a finite set of exceptional graphs. A recent result of Zhang and Wu (2024) [36] implies that the conjecture holds for every connected triangle-free graph that is not a 4-cycle. In this paper, we prove this conjecture. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For a fixed graph H, a graph G is called H-saturated if G does not contain H as a (not necessarily induced) subgraph, but G + e contains a copy of H for any e is an element of E(G & strns;). The saturation number of H, denoted by sat(n, H), is the minimum number of edges in an n-vertex H-saturated graph. A wheel Wn is a graph obtained from a cycle of length n by adding a new vertex and joining it to every vertex of the cycle. A well-known result of Erdos, Hajnal and Moon shows that sat(n, W-3) = 2n - 3 for all n >= 4 and K-2 boolean OR K & strns;(n-2) is the unique extremal graph, where boolean OR denotes the graph join operation. In this paper, we study the saturation number of W-4. We prove that sat(n, W-4) = & LeftFloor;(5n-10)/(2) & RightFloor; for all n >= 6 and give a complete characterization of the extremal graphs. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For a fixed graph H, a graph G is H-saturated if G does not contain a copy of H, but adding any edge e ∈ E(G) to G creates a copy of H. The saturation number sat(n,H) is the minimum number of edges in an H-saturated graph on n vertices. Let K be the kite graph, formed by removing one edge from K_4 and then attaching a pendant edge to a vertex of degree two in the resulting graph.In this paper, we first establish a relationship between connectivity and K-saturated graphs, and subsequently determine the saturation number of the kite graph K. Moreover, we completely characterize all extremal graphs.Our result provides a partial answer to a problem raised by Hua and Peng [Discrete Math. 349 (2026) 114674].

Let G be a graph and F be a family of graphs. We say a graph G is F-saturated if G does not contain any member in F and for any e is an element of E(G), G + e creates a copy of some member in F. The saturation number of F is the minimum number of edges of an F-saturated graphs of n vertices, denoted by sat(n, F). If F = {F}, then we write it as sat(n, F) for short. In this paper, we determine the exact value of sat(n, {K3, Pk}), and as its application, we obtain two bounds on sat(n, K3 boolean OR Pk) for k >= 10 and sufficiently large n. Furthermore, sat(n, K1 boolean OR F) is determined, where F is a linear forest without isolated vertices. (c) 2025 Published by Elsevier B.V.

For a fixed graph F, a graph G is F-saturated if G does not contain F as a subgraph, but adding any edge in E(G) will result in a copy of F. The minimum size of an F-saturated graph of order n is called the saturation number of F, denoted by sat(n, F). In this paper, we are interested in saturation problem of graph K1 boolean OR Pt for t >= 2. As some known results, sat(n,K1 boolean OR Pt) is determined for 2 <= t <= 4. We will show that sat(n,K1 boolean OR Pt)=(n-1)+sat(n-1,Pt) for t >= 5 and n sufficiently large. Moreover, (K1 boolean OR Pt)-saturated graphs with sat(n,K1 boolean OR Pt) edges are characterized.

For any positive integer k and any graph G, a subset D of vertices of G is called a k-clique isolating set of G if G - N[D] does not contain k-clique as a subgraph. The k-clique isolation number of G, denoted by iota(G,k), is the minimum cardinality of a k-clique isolating set of G. Borg, Fenech and Kaemawichanurat (Discrete Math. 343 (2020) 111879) proved that if G is a connected n-vertex graph, then iota(G, k) <= n k+1 unless G is a k-clique, or k = 2 and G is a 5-cycle. At the end of their paper, Borg, Fenech and Kaemawichanurat asked for a characterization of all connected n-vertex graphs G such that iota(G, k) = n k+1 . An old result of Payan and Xuong, and independently of Fink et al., in the 1980s has already answered this problem for the case k = 1. Very recently, the case when k = 2 was solved by Boyer and Goddard, and the case when k = 3 was solved by the first two authors of the present paper and Zhang. In this paper, we solve all the remaining cases. We show that except an infinite family of graphs, there are exactly 7 such graphs when k = 4 and exactly k+ 2 such graphs when k >= 5. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For a fixed graph F, a graph G is F-saturated if G does not contain F as a subgraph, but adding any edge in E(G) will result in a copy of F. The minimum size of an F-saturated graph of order n is called the saturation number of F, denoted by sat(n, F). In this paper, we are interested in saturation problem of graph K_1∨P_t for t≥ 2 . As some known results, sat(n,K_1∨P_t) is determined for 2≤ t≤ 4 . We will show that sat(n,K_1∨P_t)=(n-1)+sat(n-1,P_t) for t≥ 5 and n sufficiently large. Moreover, (K_1∨P_t) -saturated graphs with sat(n,K_1∨P_t) edges are characterized.

For any graph G, let gamma(G) and i(G) denote the domination number and the independent domination number of G, respectively. For any positive integer k, a subset S of vertices in a graph G is said to be a k-independent set of G if G[S] has maximum degree less than k. The k-independence number of G, denoted by alpha(k)(G), is the maximum cardinality of a k-independent set of G. Let T be any tree with n >= 2 vertices. Dehgardi et al. proved that i(T) <= 3/4 alpha(2)(T) and gamma(T) + i(T) <= 4/3 alpha(2)(T). Later, Zhang and Wu extended the former result of Dehgardi et al. by showing that i(T) <= k+1/2k alpha(k)(T), and conjectured that the latter one can also be generalized to gamma(T) + i(T) <= 2k/2k-1 alpha(k)(T). In this paper, we prove this conjecture, and moreover, we characterize all extremal trees for which the equality holds. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A subset D of vertices in a graph G is a dominating set if every vertex in V(G)∖ D is adjacent to at least one vertex in D. For any positive integer k, a subset S of vertices in G is a k-independent set if G[S] has maximum degree less than k. The k-independence number of G, denoted by α _k(G) , is the maximum cardinality of a k-independent set in G. A subset I of vertices in G is a k-independent dominating set if I is both k-independent and dominating. The k-independent domination number of G, denoted by i_k(G) , is the minimum cardinality of a k-independent domination set in G. Recently, Zhang and Wu (J Oper Res Soc China 12:485–494, 2024) showed that i_2(T)⩽2/3α _2(T) for any nontrivial tree T, and this bound is sharp. In this paper, we give a complete characterization of all trees attaining this bound, which resolves a problem proposed by Zhang and Wu. Moreover, we further prove that i_3(T)⩽3/5α _3(T) for any nontrivial tree T, and characterize all extremal trees for which the equality holds.

For any graph G, a subset D of vertices of G is called a cycle isolating set of G if G - N[D] contains no cycle. The cycle isolation number of G, denoted by tc(G), is the minimum cardinality of a cycle isolating set of G. Analogously, for any positive integer k and any graph G, a subset D of vertices of G is called a Kk-isolating set of G if G - N[D] does not contain Kk as a subgraph. The Kk-isolation number of G, denoted by t(G, Kk), is the minimum cardinality of a Kk-isolating set of G. Borg (2020) proved that if G is a connected n-vertex graph and G not congruent to C3, then tc(G) <= n4. Borg, Fenech and Kaemawichanurat (2020) showed that if G is a connected n-vertex graph, then t(G, Kk)<= n k+1 unless G similar to= Kk or k = 2 and G similar to= C5. These settled two problems of Caro and Hansberg (2017), and both bounds are sharp. Borg, Fenech and Kaemawichanurat (2020) further asked for a characterization of all connected n-vertex graphs G such that t(G, Kk) = k+1 n . The case when k = 2 for this problem was recently solved by Boyer and Goddard (2024). In this paper, we characterize all connected n-vertex graphs G such that tc(G) = n4. As a consequence, we also characterize all connected n-vertex graphs G such that t(G, K3) = n4. This solves the aforementioned problem of Borg, Fenech and Kaemawichanurat (2020) for the case k = 3. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

An injective edge-coloring of a graph G is an edge-coloring of G such that any two edges that are at distance 2 or in a common triangle receive distinct colors. The injective chromatic index of G is the minimum number of colors needed to guarantee that G admits an injective edge-coloring. Ferdjallah, Kerdjoudj and Raspaud showed that the injective chromatic index of every subcubic graph is at most 8, and conjectured that 8 can be improved to 6. Kostochka, Raspaud and Xu further proved that every subcubic graph has the injective chromatic index at most 7, and every subcubic planar graph has the injective chromatic index at most 6. In this paper, we consider the injective edge-coloring of claw-free subcubic graphs. We show that every connected claw-free subcubic graph, apart from two exceptions, has the injective chromatic index at most 5. We also consider the list version of injective edge-coloring and prove that the list injective chromatic index of every claw-free subcubic graph is at most 6. Both results are sharp and strengthen a recent result of Yang and Wu which asserts that every claw-free subcubic graph has the injective chromatic index at most 6.

For any non-negative integer k and any graph G, a subset S⊆ V(G) is said to be a K_1,k+1 -isolating set of G if G-N[S] does not contain K_1,k+1 as a subgraph. The K_1,k+1 -isolation number of G, denoted by ι _k(G) , is the minimum cardinality of a K_1,k+1 -isolating set of G. Recently, Zhang and Wu (2021) proved that if G is a connected n-vertex graph and G∉{P_3,C_3,C_6} , then ι _1(G)≤2/7n . In this paper, we characterize all extremal graphs attaining this bound, which resolves a problem proposed by Zhang and Wu (Discrete Appl Math 304:365–374, 2021).

For any graph G , a subset S of vertices of G is said to be a cycle isolating set of G if G-N_G[S] contains no cycle, where N_G[S] is the closed neighborhood of S . The cycle isolation number of G , denoted by ι _c(G) , is the minimum cardinality of a cycle isolating set of G . Recently, Borg (2020) showed that if G is a connected n -vertex graph that is not isomorphic to C_3 , then ι _c(G)≤n/4 . In this paper, we present a sharp upper bound on the cycle isolation number of a connected graph in terms of its number of edges. We prove that if G is a connected m -edge graph that is not isomorphic to C_3 , then ι _c(G)≤m+1/5 . Moreover, we characterize all connected graphs attaining this bound.

A strong edge-coloring of a graph G is an edge-coloring of G such that any two edges that are either adjacent to each other or adjacent to a common edge receive distinct colors. The strong chromatic index of G, denoted by χ '_s(G) , is the minimum number of colors needed to guarantee that G admits a strong edge-coloring. For any integer n≥ 3 , let H_n denote the n-prism (i.e., the Cartesian product C_n□ K_2 ) and H_n^Δ the graph obtained from H_n by replacing each vertex with a triangle. Recently, Lin and Lin (2022) asked whether χ '_s(H_n^Δ)=6 for any n≥ 3 . In this short note, we answer this question in the affirmative.

An even cycle decomposition of a graph is a partition of its edges into even cycles. Markström constructed infinitely many 2-connected 4-regular graphs without even cycle decompositions. Máčajová and Mazák then constructed an infinite family of 3-connected 4-regular graphs without even cycle decompositions. In this note, we further show that there exists an infinite family of 4-connected 4-regular graphs without even cycle decompositions.

A star edge-coloring of a multigraph G is a proper edge-coloring of G such that no path or cycle of length four is bi-colored. The star chromatic index of G is the minimum number of colors needed to guarantee that G admits a star edge-coloring. The list star chromatic index of G is the smallest integer k such that for any k-uniform list assignment L for the set of edges, G has a star edge-coloring from L. Dvořák, Mohar and Šámal proved that every subcubic multigraph has star chromatic index at most 7, and conjectured that 7 can be further improved to 6. Lužar, Mockovčiaková and Soták strengthened the result of Dvořák, Mohar and Šámal by showing that every subcubic multigraph has list star chromatic index at most 7. In this paper, we verify the conjecture of Dvořák, Mohar and Šámal for a class of subcubic multigraphs. We prove that every claw-free subcubic multigraph has list star chromatic index at most 6, and give a few examples to show that the upper bound is tight.

A graph is said to be ISK4 ‐free if it does not contain any subdivision of K4 as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ISK4 ‐free graph is 4‐colorable. In this paper, we show that this conjecture is true for the class of { ISK4 , diamond, bowtie}‐free graphs, where a diamond is the graph obtained from K4 by removing one edge and a bowtie is the graph consisting of two triangles with one vertex identified.