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A note on the edge general position number of cactus graphs

  • Yahan Cao ORCID logo and Shengjin Ji ORCID logo EMAIL logo
Published/Copyright: November 17, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

For a given graph G, a subset S of E(G) is an edge general position set of G if no triple of S is contained in a common shortest path. The cardinality of a largest edge general position set of G is called the edge general position number of G, denoted by gp e (G). In the paper, sharp upper and lower bounds of the edge general position number are obtained among cactus graphs. Moreover, we characterize all graphs that attained these bounds.

Keywords: general position set; edge general position set; cut vertex; cactus graph
MSC 2020: 05C12; 05C35; 05C70

1 Research background

Let G = (V(G), E(G)) be a finite simple graph with vertex set V(G) and edge set E(G). As usual, |V(G)| and |E(G)| are called the order and the size of G, respectively. If uv ∈ E(G), then we say that u is a neighbor of v in G and vice versa. For a vertex v ∈ V(G), set N G (v) = {u ∈ V(G) | uv ∈ E(G)} is regarded as the open neighborhood of v. The degree of a vertex v ∈ V(G) is d G (v) = |N G (v)|. The general position problem in graph theory is to find a largest set of vertices S ⊆ V(G), called a gp-set of G, such that no shortest path of G contains three vertices of S, which was first proposed by Manuel and Klavžar [1]. The general position number (gp-number for short) of G, denoted by gp(G), is the cardinality of a gp-set of G. In fact, they researched the basic properties and the bounds of gp-number in some special graphs, meanwhile, they proved that the general position problem is NP-complete. Note that the classical general position problem is traced back close to the celebrated century-old problem named as the no-three-in-line problem, first introduced by Dudeney [2] in 1917. Recently, Payne and Wood [3] extended the no-three-in-line problem to the general position subset selection problem in discrete geometry. For progress in this regard, see [4],5] and references therein.

Now we focus on the general position problem in graph theory. Patkós [6] studied the gp-number of Kneser graphs, and determined the exact value of gp-number of some special Kneser graphs by using a generalization of Bollobás’s inequality on intersecting set pair systems. Klavžar et al. [7] and Tian and Xu [8] studied the gp-number of Cartesian products. Tian et al. [9] determined the gp-numbers of maximal outerplanar graphs. For further results, see the recent survey on gp-number [10].

The edge general position set of a graph is the edge version of the general position set of a graph, see the seminal paper [11]. We now introduce formally its definition as follows. Let G = (V(G), E(G)) be a graph and S ⊆ E(G). We say that S is an edge general position set if no three edges of S lie on a common shortest path. The subset S is also called a maximum edge general position set if it has the largest cardinality in all edge general position sets. We also call S a gp e -set of G for short. The edge general position number (gp e -number for short) of G, denoted by gp e (G), is the cardinality of a gp e -set in G. An edge general position problem is to find a gp e -set of graphs. And then it should be added that the edge general position set has been recently extended to k-edge general position sets in [12]. Klavžar and Tan [13] obtained the sharp bounds of gp e -number on Fibonacci and Lucas Cubes. Note that many researchers concerned the extremal problems of cactus graphs, cf. [14], [15], [16], [17]. Hence, it is interesting to study the gp e -number of cactus graphs. In the paper, we continue the research in this direction.

For convenience, we now introduce some notations. A block of G is a maximal connected subgraph of G without cut vertex. A connected graph G is called a cactus graph if its any block is either a cycle or an edge. Let C n k be the set of all cactus graphs of order n with k ≥ 1 cycles. Let C n k , t be the set of cactus graphs on order n with k cycles and t leaves, where t ≥ 0 and k ≥ 1.

Let G ∈ C n k , t be a graph. A cyclic path is a path connecting two cycles in G such that except two end vertices, any internal vertex (if it exists) does not belong to any cycle of G. Note that two cyclic paths can overlap, and internal vertices of a cyclic path do not necessarily have degree 2. As shown in Figure 1, (x, y)-path, (u, v)-path, (u, w)-path and (w, v)-path are cyclic paths. We use P ( G ) to denote the set of cyclic paths of G. Let P and C be a cyclic path in G and a cycle of G, respectively. For a cut vertex u of G contained in V(C) ∪ V(P), by removing all edges in E(C) ∪ E(P) incident with u, the component containing u, denoted by T u , is named as a root tree of G with root u if T u does not contain cycles. A root tree of G is a tree with roots on a cycle or a cyclic path. We use T ( G ) to denote the set of all root trees of G. Moreover, a vertex w of a root tree is called a leaf if w has degree one and is not the root. Clearly, the number of leaves in T ( G ) equals t. A vertex v ∈ V(G) is called a cut vertex if removing v increases the number of connected components. An inner cut vertex is a vertex of a cycle shared by another cycle or a cyclic path. Clearly, an inner cut vertex is indeed a cut vertex. We use c(C) to denote the number of cut vertices of C and p e (C) to denote the number of pendant edges of root trees on C such that they are not shared by other cycles or cyclic paths.

Figure 1: The two examples of cyclic paths.
Figure 1:

The two examples of cyclic paths.

In the paper, sharp upper and lower bounds of gp e -number are obtained among cactus graphs with k cycles and t pendant leaves. Moreover, we characterize the structures of these graphs that attain the bounds.

2 Edge general position sets of cactus graphs

In the section, we first give some notations which will be useful in showing our main results. We will obtain the upper bound of gp e -number in C n k , t and characterize the graphs attaining the bound. And then, we will show the lower bounds of gp e -number in cactus graphs, meanwhile, the extremal graphs are obtained completely.

2.1 Notations

2.1.1 An inner cycle and an outer cycle

We now define several types of cycles of G by means of the cut vertices contained in these cycles. A cycle C l is an inner cycle if there are at least two subgraphs of G − E(C l ) containing cycles, an outer cycle otherwise. In particular, an outer cycle with exactly one cut vertex is an end-block.

Figure 2: A root chain cactus graph.
Figure 2:

A root chain cactus graph.

2.1.2 A chain cactus and a root chain cactus

A chain cactus G is a cactus graph in which all blocks have at most two cut vertices, and each cut vertex is exactly shared by two blocks. Clearly, a chain graph G has exactly two outer cycles and at most two leaves. We call a graph G′ a root chain cactus if it is obtained from G by changing at least one outer cycle of G such that it contains exactly one root tree with two leaves at a vertex other than the inner cut vertex, see Figure 2. The subgraph of G formed from two outer cycles and the inner cycles and cyclic paths connecting them is called a subchain cactus of G. A subchain cactus of the chain cactus graph presented in Figure 3 is obtained from it by removing two leaves.

Figure 3: A chain cactus graph.
Figure 3:

A chain cactus graph.

2.1.3 A cut-path of C l and D c (C l )

Let u i and u j be two vertices of C l , clearly, C l can be regarded as consisting of two (u i , u j )-paths. If all cut vertices lying on C l belong to one (u i , u j )-path, then the path is referred to as a (u i , u j )-cut-path (or a cut-path for short) and then denote d c (u i , u j ) the number of edges contained in it, e.g., d c (z 1, z 3) = 4 see Figure 4. In particular, suppose now that a cycle C l of G has at least three cut vertices. If there are three cut vertices satisfying the following: two cut vertices x i and x j form a (x i , x j )-cut-path only containing the third cut vertex x k while it is the root vertex of a root tree T x k with a leaf, then we name it a (x i , x j )-root-cut-path (or a root-cut-path for short) and denote the number of edges on the path by d r (x i , x j ). And then the vertex u is referred to as a bad vertex if either d r ( x i , x j ) ≥ l 2 + 1 for even l or d r ( x i , x j ) ≥ ⌊ l 2 ⌋ + 1 otherwise. As shown in Figure 4, d r (x 1, x 3) = 4 and x 2 is a bad vertex. Set D c ( C l ) = min u i , u j ∈ V ( C l ) d c ( u i , u j ) (D c for short), e.g., D c (C 4) = 3, see Figure 4.

Figure 4: Illustrative examples of cut-path, root cut-path and D c .
Figure 4:

Illustrative examples of cut-path, root cut-path and D c .

2.1.4 A good cycle, a normal cycle and a bad cycle

We now give a classification of cycles in G under the assumption that if C l is an outer cycle then c(C l ) ≥ 4. We call the cycle C l a normal I cycle if D c is no more than l 2 − 1 for even l or ⌊ l 2 ⌋ for odd l. Conversely, assume now that D c ≥ l 2 for even l or D c ≥ ⌊ l 2 ⌋ + 1 for odd l. The cycle C l is named as either a normal II cycle if there exists a root-cut-path having a bad vertex or a bad cycle otherwise. For example, C 3 is a normal II cycle, see Figure 5.

Figure 5: Illustrative examples of different kinds of cycles.
Figure 5:

Illustrative examples of different kinds of cycles.

In particular, suppose next that C l is an outer cycle with c(C l ) ≤ 3. For c(C l ) = 1, clearly, C l is an end-block. Then C l is either a normal I cycle for even order, or a good cycle otherwise. Assume now that 2 ≤ c(C l ) ≤ 3.

Let the order of C l be even. The cycle C l is a normal I cycle with the condition D c ≤ l 2 − 1 . Under the condition D c ≥ l 2 + 1 , C l is a normal II cycle if it contains a root-cut-path with a bad vertex or a bad cycle otherwise. For the case D c = l 2 and c(C l ) = 3, we call that C l is a normal II cycle if there exists a root-cut-path with a bad vertex, a bad cycle otherwise, such as, a normal II cycle C 1 and a bad cycle C 4 of G 4 shown in Figure 5. For the case D c = l 2 and c(C l ) = 2, we call C l either a bad cycle if p e (C l ) ≥ 2 or a normal II cycle otherwise. We can check that C 6 of G 4 is a bad cycle, see Figure 5.

Suppose the order of C l is odd. The cycle C l is a normal I cycle if D c ≤ ⌊ l 2 ⌋ . Moreover, C l with D c ≥ ⌊ l 2 ⌋ + 1 and c(C l ) = 3 is a normal II cycle if a root-cut-path of C l has a bad vertex, a bad cycle otherwise.

2.2 The upper bounds of the edge general position number

From [18], we know the following proposition.

Proposition 2.1.

gp e (C n ) = n if n ∈ {3, 4, 5}, and gp e (C n ) = 4 otherwise.

Observation 2.1.

For arbitrary graph G ∈ C n k , t with k ≥ 2, let T ( G ) be the set of root trees with t leaves in G, then there exists a gp e -set S such that | S ∩ E ( T ( G ) ) | ≤ t .

Lemma 2.1.

Let G ∈ C n k , t be a graph with k ≥ 2 and P ( G ) be the set of cyclic paths of G. Then there exists a gp e -set S such that | S ∩ E ( P ( G ) ) | = 0 .

Proof.

Suppose G ∈ C n k , t is a cactus graph. Assume that S is a gp e -set of G having edges from outer cycles and T ( G ) as more as possible. Let P ( G ) be the set of cyclic paths of G and an edge e ∈ E ( P ( G ) ) . By contradiction, assume that S is a gp e -set of G with e ∈ S. Let H 1 and H 2 be two components of G − {e}. Observe that H i contains at least one outer cycle from the definition of a cyclic path, and every shortest path from V(H 1) to V(H 2) goes through e. In addition, if C is an end-block of G, then we observe that either |S ∩ E(C)| ≤ 3 for odd order of C or |S ∩ E(C)| ≤ 2 for even order of C. Based on the types of outer cycles, we will take three cases to proceed the proof.

Case 1 H 1 and H 2 contain end-blocks.

Let C i be an end-block of H i and u i be the unique cut vertex for i = 1, 2. We first consider that the lengths of C 1 and C 2 have the same parity. If |V(C 1)| and |V(C 2)| are odd, then they are good cycles. From the maximum of S and e ∈ S, we deduce that one of |S ∩ E(C 1)| and |S ∩ E(C 2)| equals 1 and the other equals 3. Without loss of generality, assume that |S ∩ E(C 1)| = 1 and |S ∩ E(C 2)| = 3. Let S 1 = (S − {e}) ∪ {e 1, e 2}, where e 1 and e 2 are two edges incident with u 1 in C 1, as shown in Figure 6. It follows that S 1 is a new edge general position set of G larger than S, a contradiction. If |V(C 1)| and |V(C 2)| are even, then they are normal I cycles. By the choice of S, we obtain that one of |S ∩ E(C 1)| and |S ∩ E(C 2)| equals 0 and the other equals 2. Without loss of generality, assume that |S ∩ E(C 1)| = 0 and |S ∩ E(C 2)| = 2. Let S 2 = (S − {e}) ∪ {e 1, e 2}, where e 1 and e 2 are two edges incident with u 1 in C 1, as shown in Figure 7. Obviously, |S 2| > |S|. In addition, observe that S 2 is also an edge general position set of G, contradicting our assumption. We now assume that the lengths of C 1 and C 2 have different parity. Using the similar way of the first case, we also have done. So we omit the process here.

Case 2 H 1 and H 2 contain no end-blocks.

Assume that C 1 ′ and C 2 ′ are two outer cycles of H 1 and H 2, respectively. Let u i be an inner cut vertex of C i ′ for i = 1, 2. Since they are not end-blocks, C i ′ contains at least one root of some root tree for i = 1, 2. Let L i be the set of pendant edges belonging to root trees on C i for i = 1, 2. From the maximum of S and e ∈ S, we conclude that at least one of S ∩ L 1 = ∅ and S ∩ L 2 = ∅ is valid.

We now consider that S ∩ L 1 = ∅ and S ∩ L 2 = ∅ hold simultaneously. So we deduce that | S ∩ E C i ′ | ∈ { 2,3 } according to the parity of the lengths of these two cycles. In particular, if | S ∩ E C 1 ′ | = | S ∩ E C 2 ′ | = 3 , then we will find three edges of S lying a shortest path, a contradiction. If there is one cycle, say C 1 ′ , such that | S ∩ E C 1 ′ | = 2 , then we deduce that C 1 ′ has exactly two cut vertices with D c C 1 ′ = | C 1 ′ | 2 . So there is a contradiction again. We thus assume that one of S ∩ L 1 = ∅ and S ∩ L 2 = ∅ is correct, say S ∩ L 1 = ∅ and S ∩ L 2 ≠ ∅. Hence, | S ∩ E C 1 ′ | ∈ { 2,3 } and |S ∩ E(L 2)| ≥ 2. Using the similar argument of the first case, we obtain that S contains three edges lying on a shortest path through e of G. We thus get a contradiction.

Case 3 One of H 1 and H 2 contains end-blocks.

Assume that H 1 contains an end-block, denoted by C 3. Let u 3 be the inner cut vertex in C 3. Then C 3 is a normal I cycle for even order or a good cycle otherwise. Let C 4 be an outer cycle in H 2 and L 3 be the set of pendant edges belonging to root trees of C 4. Then one of S ∩ L 3 = ∅ and S ∩ L 3 ≠ ∅ holds. For each case, the choice of S and e ∈ S imply that |S ∩ C 3| ∈ {0, 1}. Let S 3 = (S − {e}) ∪ {e 31, e 32} with |S 3| > |S|, where e 31 and e 32 incident with u 3 of C 3. It follows that S 3 is also an edge general position set of G, a contradiction.

Therefore, we finish the proof.□

Figure 6: Used to illustrate Case 1, both C 1 and C 2 are odd cycles.
Figure 6:

Used to illustrate Case 1, both C 1 and C 2 are odd cycles.

Figure 7: Used to illustrate Case 1, both C 1 and C 2 are even cycles.
Figure 7:

Used to illustrate Case 1, both C 1 and C 2 are even cycles.

Lemma 2.2.

Suppose G ∈ C n k , t with k ≥ 2 cycles. Let C 0 be a cycle of G. Then there exists a gp e -set S such that |E(C 0) ∩ S| ≤ 3 and |E(C 0) ∩ S| ∈ {0, 2, 3}. In addition, the following assertions hold.

  • (i) |E(C 0) ∩ S| = 3 if and only if C 0 is a good cycle.

  • (ii) |E(C 0) ∩ S| = 2 if and only if C 0 is a normal I cycle or a normal II cycle.

  • (iii) |E(C 0) ∩ S| = 0 if and only if C 0 is a bad cycle.

Proof.

Let G be a cactus graph with k ≥ 2 cycles and t leaves. Let S be a gp e -set containing pendant edges and these edges from outer cycles as more as possible. By means of Lemma 2.1, the elements of S are derived from cycles and root trees of G. Assume that C 0 is a cycle of G. We first prove the following claim.□

Claim 1. |S ∩ E(C 0)| ≤ 3.

Proof.

By contradiction, suppose |S ∩ E(C 0)| ≥ 4. Together with Proposition 2.1, we conclude that |S| = |S ∩ E(C 0)| ∈ {4, 5}.

If C 0 is an inner cycle, then we deduce that G is a chain cactus by the maximum of S. Assume that C 1 and C 2 are two outer cycles of G with two inner cut vertices u 1 and u 2, respectively. In addition, by the choice of S, |S ∩ E(C i )| = 0 for i = 1, 2. Clearly, C 0 is an even cycle. Otherwise, D c ( C 0 ) ≤ ⌊ | C 0 | 2 ⌋ . Then by choosing the two end edges of cut-path of C 0 with length |C 0| − D c (C 0), say e 0, e 0 ′ , we get a new edge set S′ from S by removing all edges from C 0 and adding two edges e 0 , e 0 ′ and four edges from C 1 and C 2 respectively incident with u 1 and u 2. Evidently, S′ is an edge general position set of G with |S′| > |S|, we get a contradiction with the maximum of S. Hence, |S| = 4, which infers that C 1 and C 2 are even cycles. We thus deduce that D c ( C 0 ) = | C 0 | 2 . For each C i , if C i has two cut vertices, then D c ( C i ) = | C i | 2 , which contradicts the choice of S.

If C 0 is an outer cycle with the inner cut vertex u 0, then by the choice of S we have that G is a chain cactus graph. In other words, all inner cycles are bad. Let C 0 ′ be another outer cycle with an inner cut vertex u 0 ′ . But we find that | S ∩ E C 0 ′ | = 0 , which contradicts the choice of S.

Hence, |S ∩ E(C 0)| ≤ 3 is true.□

We now divide the following cases to finish the remaining proof.

Case 1 C 0 is an inner cycle of G.

In the case, we know that k ≥ 3. Note that, for each inner cycle C 0, there are two outer cycles C 1 and C 2 of G for which C 0 is lying on its unique subchain cactus between C 1 and C 2. Let u 1 and u 2 be the two inner cut vertices of C 0 belonging to the subchain cactus. By the maximum of S, we can claim that |S ∩ E(C 0)| ≤ 2. Assume to the contrary that |S ∩ E(C 0)| ≥ 3. Recall that |S ∩ E(C 0)| ≤ 3. So |S ∩ E(C 0)| = 3. Set {e 1, e 2, e 3} ⊆ S ∩ E(C 0). So there exists an edge, say e 1, such that it lies on a (u 1, u 2)-path with length no more than | C 0 | 2 − 1 for even order (or | C 0 | − 1 2 for odd order). In addition, the choice of S implies that one of |S ∩ E(C 1)| and |S ∩ E(C 2)| equals zero, say |S ∩ E(C 2)| = 0. Set S′ = (S − e 1) ∪ {e 4, e 5}, where e 4, e 5 ∈ E(C 2) with the same distance to e 1, as shown in Figure 8. Evidently, S′ is an edge general position set with larger size than that of S, a contradiction.

Figure 8: Used to illustrate Case 1.
Figure 8:

Used to illustrate Case 1.

Observe that, if D c ( C 0 ) ≤ | C 0 | 2 − 1 for even order (or | C 0 | − 1 2 for odd order), we obtain that |S ∩ E(C 0)| = 2. In fact, there is a (w 1, w 2)-cut-path with length D c (C 0) in C 0. Conversely, for another (w 1, w 2)-path of C 0, we can choose its two end edges as the elements of S. Thus, C 0 is a normal I cycle. Furthermore, assume that D c ( C 0 ) ≥ | C 0 | 2 for even order (or | C 0 | − 1 2 + 1 for odd order). If there is a bad vertex u for which a root-cut-path containing u has length greater than | C 0 | 2 + 1 for even order (or | C 0 | − 1 2 + 1 for odd order). Then |S ∩ E(C 0)| = 2 and C 0 is a normal II cycle. Otherwise, |S ∩ E(C 0)| = 0 and C 0 is a bad cycle. Hence, in the case, |E(C 0) ∩ S| ≠ 1.

Case 2 C 0 is an outer cycle of G.

Note that c(C 0) ≥ 1. Assume now that c(C 0) = 1, which implies that C 0 is an end-block. The choice of S results in either |S ∩ E(C 0)| = 3 for odd order or |S ∩ E(C 0)| = 2 for even order.

We now assume that c(C 0) ≥ 4. According to the values of D c (C 0), we conclude that |S ∩ E(C 0)| = 2 with D c ( C 0 ) ≤ | C 0 | 2 − 1 for even order (or | C 0 | − 1 2 for odd order). Thus, C 0 is a normal I cycle. Suppose that D c ( C 0 ) ≥ | C 0 | 2 for even order (or | C 0 | − 1 2 + 1 for odd order). If there is a bad vertex u such that a root-cut-path containing u has the length no less than | C 0 | 2 + 1 for even order (or | C 0 | − 1 2 + 1 for odd order). Then |S ∩ E(C 0)| = 2 and C 0 is a normal II cycle. Otherwise, |S ∩ E(C 0)| = 0 and C 0 is a bad cycle.

Assume next that c(C 0) = 3. By the values of D c (C 0), we deduce that |S ∩ E(C 0)| = 2 with D c ( C 0 ) ≤ | C 0 | 2 − 1 for even order (or | C 0 | − 1 2 for odd order) and C 0 is a normal I cycle. Suppose now that D c ( C 0 ) ≥ | C 0 | 2 for even order (or | C 0 | − 1 2 + 1 for odd order). If there is a root-cut-path with a bad vertex, then |S ∩ E(C 0)| = 2 and C 0 is a normal II cycle. Otherwise, |S ∩ E(C 0)| = 0 by the choice of S, so C 0 is a bad cycle.

We now consider the case c(C 0) = 2. It is clear that D c ( C 0 ) ≤ | C 0 | 2 for even order (or | C 0 | − 1 2 for odd order). If C 0 is an odd cycle, then we deduce that |S ∩ E(C 0)| = 2 and C 0 is a normal I cycle, as shown in Figure 9. Assume that C 0 is an even cycle. We can verify that, if D c ( C 0 ) ≤ | C 0 | 2 − 1 , then |S ∩ E(C 0)| = 2 and C 0 is a normal I cycle, as shown in Figure 9. Otherwise D c ( C 0 ) ≥ | C 0 | 2 . We deduce that C 0 is either a normal II cycle with |S ∩ E(C 0)| = 2 and p e (C 0) = 1 or a bad cycle with |S ∩ E(C 0)| = 0 and p e (C 0) ≥ 2, as shown in Figures 10 and 11. Clearly, in the case, |E(C 0) ∩ S| ≠ 1.

Figure 9: Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a normal I cycle.
Figure 9:

Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a normal I cycle.

Figure 10: Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a bad cycle.
Figure 10:

Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a bad cycle.

Figure 11: Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a normal II cycle.
Figure 11:

Used to illustrate Case 2 with c(C 0) = 2 and C 0 is a normal II cycle.

Therefore, we have done as required. □

Based on the above conclusions, we deduce the following result.

Theorem 2.1.

For k ≥ 2, let G ∈ C n k , t be a graph with r odd cycles and k − r even cycles. Then gp e (G) ≤ 2(k − r) + 3r + t with equality only if all odd cycles are good and all even cycles are normal I.

Proof.

Suppose that G ∈ C n k , t is a graph with r odd cycles and k − r even cycles and let S be a gp e -set of G containing as many pendant edges and edges from end-blocks as possible. Let C l be a cycle of G with length l. Combining Lemmas 2.1, 2.2 and Observation 2.1, we obtain that

| S | ≤ 2 ( k − r ) + 3 r + t .

Hence, we next show the second part of the conclusion. Assume now that G is a cactus graph such that gp e (G) = |S| = 2(k − r) + 3r + t. From Lemma 2.2, we have that |S ∩ E(C l )| ≤ 3, and then, |S ∩ E(C l )| ≤ 2 for even l. We first claim that each odd cycle C l is an end-block. If it is not an end-block, then |S ∩ E(C l )| ≤ 2 by Lemma 2.2. So we get that.

| S | = | S ∩ E ( C l ) | + | S ∩ ( E ( G ) − E ( C l ) ) | ≤ 2 + | S ∩ ( E ( G ) − E ( C l ) ) | ≤ 2 + 2 ( k − r ) + 3 ( r − 1 ) + t < 2 ( k − r ) + 3 r + t ,

a contradiction. We next claim that each even cycle C l is a normal I cycle. Contrary to our claim, suppose that C l is not normal I. Hence, C l is either a normal II cycle or a bad cycle. From by Lemma 2.2, we deduce that either |S| ≤ ≤ 2 + 2(k − r) + 3(r − 1) + t − 1 < 2(k − r) + 3r + t for the first case, or |S| ≤ ≤ 0 + 2(k − r) + 3(r − 1) + t < 2(k − r) + 3r + t otherwise. We get a contradiction.

Combining the above two cases, |S| = 2(k − r) + 3r + t implies that G contains r good odd cycle and k − r even normal I cycle. In other words, all root trees are lying on cut-paths with length less than a half of the order of each even cycle. Consequently, | S ∩ E ( T ( G ) ) | = t . Therefore, we verify the conclusion as claimed.□

The following two results hold directly from Theorem 2.1.

Corollary 2.2.

Let G ∈ C n k , t be a graph with k ≥ 2, then gp e (G) ≤ 3k + t with equality only if G has k good odd cycles.

Corollary 2.3.

Suppose that G ∈ C n k , t is a graph with k (≥2) even cycles. We have gp e (G) ≤ 2k + t with equality if and only if all even cycles of G are normal I.

Theorem 2.4.

Let G ∈ C n k , t be a graph with k ≥ 1, then

g p e G ≤ max { 5 , 3 k + t } ,

where equality holds if and only if all cycles of G are good odd cycles.

Proof.

Suppose that G ∈ C n k , t is a graph with k ≥ 1 and t ≥ 0. Let C l be a cycle of G with length l. We will consider three cases to proceed with the proof.

Case 1 k = 1 and t ≤ 1.

In fact, G is a unicyclic graph with a unique cycle C l . Clearly, 3k + t ≤ 4 < 5. It is easy to check that gp e (G) ≤ 5 with equality holds if and only if G ≅ C 5. At the time, C 5 is a good odd cycle of G.

Case 2 k = 1 and t ≥ 2.

Note that C l is a unique cycle of G. In addition, 5 ≤ 3k + t. By direct checking, we obtain that gp e (G) ≤ 3k + t with equality if and only if the length of C l is odd and C l has a unique root tree with t leaves. So C l is a good odd cycle.

Case 3 k ≥ 2.

Observe that 5 < 3k + t. Meanwhile, from Corollary 2.2, we obtain gp e (G) ≤ 3k + t with equality only if G contains k good odd cycles.□

2.3 The lower bounds of the edge general position number

Note that if a chain cactus G has two even end-blocks and its every inner cycle is bad and has two cut vertices, then gp e (G) = 4, as an example G ≅ G k see Figure 12. The graph G k also appeared in [18], Figure 2]. Observe that a cactus graph G with at least two outer cycles has gp e (G) ≥ 4. In addition, if G has t leaves, then there is an edge general position set of G consisting of t pendant edges. It follows that gp e (G) ≥ t. Observe that gp e (G k,t ) = t, where G k,t contains k − 2 triangles, 2 C 4 and t leaves such that each cycle has at least two leaves, as shown in Figure 12. Are they the lower bounds of the cactus graphs? In the following subsection, we will confirm the observations and obtain two sharp lower bounds of the cactus graphs.

Figure 12: Two examples used in Theorems 2.5 and 2.6.
Figure 12:

Two examples used in Theorems 2.5 and 2.6.

Theorem 2.5.

Let G ∈ C n k , t be a graph with k ≥ 2 and t ≥ 4. We have that gp e (G) ≥ t with equality if and only if all cycles of G are bad.

Proof.

Let G ∈ C n k , t be a graph with gp e (G) as small as possible. Recall that T ( G ) represents a set of root trees of G and L represents the set of pendant edges in T ( G ) . Let S be a gp e -set of G such that it contains as many pendant edges as possible. We observe that L is actually an edge general position set, which infers gp e (G) ≥ t. On the other hand, we can check that gp e (G k,t ) = t in which L is indeed a gp e -set, where G k,t is illustrated in Figure 6. Hence, |S| = t by the choice of G. In fact, we can claim that S = L. We assume to the contrary that |S ∩ L| ≤ t − 1.

Case 1 |S ∩ L| = t − 1.

From the assumption, we know that all outer cycles of G are not end-block. Assume that e 1 is the unique pendant edge with e 1 ∈ (L − S) and e 2 is the unique edge with e 2 ∈ (S − L). If e 1 and e 2 are lying on the same pendant path of G, then S′ = (S − e 2) ∪ {e 1} is also a gp e -set of G. We get a contradiction with the choice of S. Hence, e 2 is one edge of some cycle in G. Clearly, e 2 belongs to a normal I cycle or a normal II cycle. (Otherwise, it is contained in some outer cycle, it follows that G has a bigger edge general position set S ∪ {e 1}, a contradiction.) By Lemma 2.2, we also get a contradiction with the choice of S.

Case 2 |S ∩ L| ≤ t − 2.

Let e 1 and e 2 be two pendant edges not contained in S. From Case 1, we can assume that the edges in S that are not pendant edges lie on cycles. Let e 3 and e 4 be two elements of S not contained in L. The strategy of choosing these two edges is to make them come from the same cycle as much as possible. We can assume that e 3 and e 4 are contained in some cycle by Case 1. Hence, the cycle is either an outer cycle or an inner cycle of G, say C 0. So, from Lemma 2.2 we get a contradiction to the choice of S. Therefore, we confirm that S = L. Together with Lemma 2.2, we deduce that all cycles of G are bad.□

Theorem 2.6.

If G ∈ C n k has k ≥ 2 cycles, then gp e (G) ≥ 4 with equality only if G is either a chain cactus or a root chain cactus for which each cycle is even. In particular, each cycle with two cut vertices has D c equaling half the number of its vertices.

Proof.

Let G ∈ C n k be a graph with the minimum gp e -number. Suppose S is a gp e -set of G. Recall that gp e (G k ) = 4, so gp e (G) ≤ 4. In addition, G has at least two outer cycles, which implies that gp e (G) ≥ 4. Hence, gp e (G) = |S| = 4. Let t denote the number of leaves in G. Clearly, t ≤ 4. (Otherwise, gp e (G) ≥ 5.) Furthermore, G has exactly two even outer cycles for which each outer cycle includes at most two root trees.

Case 1 Outer cycles contain no root trees.

By our assumption, each outer cycle of G is an end-block. Hence, G is a chain cactus with two even end-blocks. Otherwise, gp e (G) ≥ 5 by Lemma 2.2, a contradiction. Evidently, S contains four proper edges from two end-blocks. Recall that |S| = 4, which infers that all inner cycles contribute 0 edge to S. By Lemma 2.2, all inner cycles are bad and even.

Case 2 An outer cycle contains a root tree, say T.

Observe that T has at most 2 leaves. We first claim that all inner cycles are bad. Otherwise, there is an inner cycle which contributes two edges to S by Lemma 2.2. Together with proper four edges in two outer cycles, we get |S| ≥ 6, a contradiction. Hence, the four edges of S are derived from the two outer cycles, where one outer cycle, denoted by C 0, has the root tree T. Mark the root of T as u′ and the inner cut vertex of C 0 as v 0. We can deduce that u′ and v 0 are diagonal of C 0.

Case 3 An outer cycle contains two root trees, denoted by T 1 and T 2.

Let C 0 be the outer cycle. Let v 1 and v 2 be the roots of T 1 and T 2, and let v 3 be the inner cut vertex of C 0. Using the same argument as in Case 1, we deduce that all inner cycles of G are bad. Observe that T 1 and T 2 are pendant paths by the minimality of S. We find that C 0 is a normal II cycle by the three cut vertices v 1, v 2 and v 3. But we can get another edge general position set with size larger than |S|, a contradiction.

Combining the above three cases, the conclusion is verified.□

3 Conclusions

As we know, the topological indices and other graph invariants have been explored on cactus graphs. In this paper, we research the edge general position number of cactus graphs, and bound it with the number of cycles and pendant vertices. Moreover, we obtain the lower bound and the upper bound by means of the number of good cycles, bad cycles and pendant vertices. We think that determining the formula of the edge general position number of cactus graphs regarding the number of cycles and pendant vertices is an interesting work in the future.

Corresponding author: Shengjin Ji, School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255000, China, E-mail: 

Acknowledgments

The authors would like to thank the editor and referees for their valuable suggestions which helps us to modify the presentation of the paper.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: Authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: Authors state no conflicts of interest.

  6. Research funding: This work was supported by the Natural Science Foundation of Shandong Province, China (No. ZR2022MA077) and by Postgraduate Education Reform Project of Shandong Province, China (No. SDYKC2023107).

  7. Data availability: Data sharing does not apply to this article as no datasets were generated or analysed during the current study.

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Received: 2024-11-11
Accepted: 2025-09-22
Published Online: 2025-11-17

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0207
Keywords for this article
general position set; edge general position set; cut vertex; cactus graph
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