Consider the following problem: Given a transitive tournament \(T\) of order \(n \geq 3\) and an integer \(k\) with \(1 \leq k \leq \binom{n}{2}\), which \(k\) ares in \(T\) should be reversed so that the resulting tournament has the largest number of spanning cycles? In this note, we solve the problem when \(7\) is sufficiently large compared to \(k\).

The bondage number \(b(G)\) of a graph \(G\) is the smallest number of edges whose removal results in a graph with domination number greater than the domination number of \(G\). Kang and Yuan [Bondage number of planar graphs. Discrete Math. \(222 (2000), 191-198]\) proved \(b(G) \leq \min\{8, \Delta + 2\}\) for every connected planar graph \(G\), where \(\Delta\) is the maximum degree of \(G\). Later Carlson and Develin [On the bondage number of planar and directed graphs. Discrete Math. \(306 (8-9) (2006), 820-826]\) presented a method to give a short proof for this result. This paper applies this technique to generalize the result of Kang and Yuan to any connected graph with crossing number less than four.

A \({Roman \;domination \;function}\) on a graph \(G = (V, E)\) is a function \(f: V(G) \to \{0, 1, 2\}\) satisfying the condition that every vertex \(u\) with \(f(u) = 0\) is adjacent to at least one vertex \(v\) with \(f(v) = 2\). The \({weight}\) of a Roman domination function \(f\) is the value \(f(V(G)) = \sum_{u \in V(G)} f(u)\). The minimum weight of a Roman dominating function on a graph \(G\) is called the \({Roman \;domination \;number}\) of \(G\), denoted by \(\gamma_R(G)\). In this paper, we study the Roman domination number of generalized Petersen graphs \(P(n, 2)\) and prove that \(\gamma_R(P(n, 2)) = \left\lceil \frac{8n}{7} \right\rceil (n\geq5)\).

Let \(G = (V, E)\) be a simple undirected graph. For an edge \(e\) of \(G\), the \({closed\; edge-neighborhood}\) of \(e\) is the set \(N[e] = \{e’ \in E \mid e’ \text{ is adjacent to } e\} \cup \{e\}\). A function \(f: E \to \{1, -1\}\) is called a signed edge domination function (SEDF) of \(G\) if \(\sum_{e’ \in N[e]} f(e’) > 1\) for every edge \(e\) of \(G\). The signed edge domination number of \(G\) is defined as \(\gamma’_s(G) = \min \left\{ \sum_{e \in E} |f(e)| \mid f \text{ is an SEDF of } G \right\}\). In this paper, we determine the signed edge domination numbers of all complete bipartite graphs \(K_{m,n}\), and therefore determine the signed domination numbers of \(K_m \times K_n\).

We discuss the primality of some corona graphs and some families of graphs.

An injective coloring of a graph \(G\) is an assignment of colors to the vertices of \(G\) so that any two vertices with a common neighbor receive distinct colors. A graph \(G\) is said to be injectively \(k\)-choosable if any list \(L(v)\) of size at least \(k\) for every vertex \(v\) allows an injective coloring \(\phi(v)\) such that \(\phi(v) \in L(v)\) for every \(v \in V(G)\). The least \(k\) for which \(G\) is injectively \(k\)-choosable is the injective choosability number of \(G\), denoted by \(\chi_i^l(G)\). In this paper, we obtain new sufficient conditions to ensure \(\chi_i^l(G) \leq \Delta(G) + 1\). We prove that if \(mad(G) \leq \frac{12k}{4k+3}\), then \(\chi_i^l(G) = \Delta(G) + 1\) where \(k = \Delta(G)\) and \(k \geq 4\). Typically, proofs using the discharging technique are different depending on maximum average degree \(mad(G)\) or maximum degree \(\Delta(G)\). The main objective of this paper is finding a function \(f(\Delta(G))\) such that \(\chi_i^l(G) \leq \Delta(G) + 1\) if \(mad(G) < f(\Delta(G))\), which can be applied to every \(\Delta(G)\).

The traditional parameter used as a measure of vulnerability of a network modeled by a graph with perfect nodes and edges that may fail is edge connectivity \(\lambda\). For the complete bipartite graph \(K_{p,q}\), where \(1 \leq p \leq q\), \(\lambda(K_{p,q}) = p\). In this case, failure of the network means that the surviving subgraph becomes disconnected upon the failure of individual edges. If, instead, failure of the network is defined to mean that the surviving subgraph has no component of order greater than or equal to some preassigned number \(k\), then the associated vulnerability parameter, the component order edge connectivity \(\lambda_c^{(k)}\), is the minimum number of edges required to fail so that the surviving subgraph is in a failure state. We determine the value of \(\lambda_c^{(k)}(K_{p,q})\) for arbitrary \(1 \leq p \leq q\) and \(4 \leq k \leq p+q\). As it happens, the situation is relatively simple when \(p\) is small and more involved when \(p\) is large.

A \(T\)-shape tree \(T(l_1, l_2, l_3)\) is obtained from three paths \(P_{l_1+1}\), \(P_{l_2+1}\), and \(P_{l_3+1}\) by identifying one of their pendent vertices. A generalized \(T\)-shape tree \(T_s(l_1, l_2, l_3)\) is obtained from \(T(l_1, l_2, l_3)\) by appending two pendent vertices to exactly \(s\) pendent vertices of \(T(l_1, l_2, l_3)\), where \(1 \leq s \leq 3\) is a positive integer. In this paper, we firstly show that the generalized \(T\)-shape tree \(T_2(l_1, l_2, l_3)\) is determined by its Laplacian spectrum. Applying similar arguments for the trees \(T_1(2l_1, l_2, l_3)\) and \(T_3(l_1, 2l_2, l_3)\), one can obtain that any generalized \(T\)-shape tree on \(n\) vertices is determined by its Laplacian spectrum.

In this paper, we use the \(q\)-difference operator and the Andrews-Askey integral to give a transformation for the Al-Salam-Carlitz polynomials. As applications, we obtain an expansion of the Carlitz identity and some other identities for Al-Salam-Carlitz
polynomials .

In this paper we define new generalizations of the Lucas numbers,which also generalize the Perrin numbers. This generalization is based on the concept of \(k\)-distance Fibonacci numbers. We give in-terpretations of these numbers with respect to special decompositions and coverings, also in graphs. Moreover, we show some identities for these numbers, which often generalize known classical relations for the Lucas numbers and the Perrin numbers. We give an application of the distance Fibonacci numbers for building the Pascal’s triangle.

This paper introduces the new notions of \(\delta-\alpha-\)open sets and the \(\delta-\alpha-\)continuous functions in the topological spaces and investigates some of their properties.

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (n_1g) \cdots (n_lg)\), where \(g \in G\) and \(n_1, \ldots, n_l \in [1, \text{ord}(g)]\), and the \({index}\) \(\text{ind}(S)\) of \(S\) is defined to be the minimum of \((n_1 + \cdots + n_l)/\text{ord}(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) when \(G = \langle g \rangle\) is a cyclic group of a prime order and \(S\) has the form \(S = g^2{(n_2g)}(n_3g){(n_4)}\). It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime order \(p \geq 31\), then every minimal zero-sum sequence \(S\) of the above-mentioned form has index \(1\), except in the case that \(S = g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)\).

The paper presents two sharp upper bounds for the largest Laplacian eigenvalue of mixed graphs in terms of the degrees and the average \(2\)-degrees, which improve and generalize the main results of Zhang and Li [Linear Algebra Appl.\(353(2002)11-20]\),Pan (Linear Algebra Appl.\(355(2002)287-295]\),respectively. Moreover, we also characterize some extreme graphs which attain these upper bounds. In last, some examples show that our bounds are improvement on some known bounds in some cases.

Cagman \(et\; al\). introduced the concept of a fuzzy parameterized fuzzy soft set(briefly, \(FPFS)\) which is an extension of a fuzzy set and a soft set. In this paper, we introduce the concepts of \(FPFS\) filters and \(FPFS\) implicative filters of lattice implication algebras and obtain some related results. Finally, we define the concept of \(FPFS\)-aggregation operator of lattice implication algebras.

We propose a practical linear time algorithm for the LONGEST PATH problem on \(2\)-trees.

By means of a \(q\)-binomial identity, we give two generalizations of Prodinger’s formula, which is equivalent to the famous Dilcher’s formula.

In this paper, we consider a random mapping \(\hat{T}_{n,\theta}\) of the finite set \(\{1,2,\ldots,n\}\) into itself, for which the digraph representation \(\hat{G}_{n,\theta}\) is constructed by: (1) selecting a random number \(\hat{L}_n\) of cyclic vertices, (2) constructing a uniform random forest of size \(n\) with the selected cyclic vertices as roots, and (3) forming `cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter \(\theta\). We investigate \(\hat{k}_{n,\theta}\), the size of a `typical’ component of \(\hat{G}_{n,\theta}\), and we obtain the asymptotic distribution of \(\hat{k}_{n,\theta}\) conditioned on \(\hat{L}_n = m(n)\). As an application of our results, we show in Section 3 that provided \(\hat{L}_n\) is of order much larger than \(\sqrt{n}\), then the joint distribution of the normalized order statistics of the component sizes of \(G_{n,\theta}\) converges to the Poisson-Dirichlet \((\theta)\) distribution as \(n \to \infty\).

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, D. S. Kim and T. Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus \((see[6])\).

Let \(H\) be a subgraph of \(G\). An \(H\)-design \((V, \mathcal{C})\) of order \(v\) and index \(\lambda\) is embedded into a \(G\)-design \((X, \mathcal{B})\) of order \(v+w\), \(w \geq 0\), and index \(\lambda\), if \(\mu \leq \lambda\), \(V \subseteq X\) and there is an injective mapping \(f: \mathcal{C} \rightarrow \mathcal{B}\) such that \(B\) is a subgraph of \(f(B)\) for every \(B \in \mathcal{C}\).
For every pair of positive integers \(v\) and \(\lambda\), we determine the minimum value of \(w\) such that there exists a balanced incomplete block design of order \(v+w\), index \(\lambda \geq 2\) and block-size \(4\) which embeds a \(K_3\)-design of order \(v\) and index \(\mu = 1\).

Let \(S\) be a finite, nonempty set of nonzero integers which contains no squares. We obtain conditions both necessary and sufficient for \(S\) to have the following property: for infinitely many primes \(p\), \(S\) is a set of quadratic nonresidues of \(p\). The conditions are expressed solely in terms of purely external (respectively, internal) combinatorial properties of the set II of all prime factors of odd multiplicity of the elements of \(S\). We also calculate by means of certain purely combinatorial parameters associated with \(\prod\) the density of the set of all primes \(p\) such that \(S\) is a set of quadratic residues of \(p\) and the density of the set of all primes \(p\) such that \(S\) is a set of quadratic nonresidues of \(p\).

For positive integers \(t\) and \(k\), the \({vertex}\) (resp. edge) Folkman number \(F_v(t,t,t;k)\) (resp. \(F_e(t,t,t;k)\)) is the smallest integer \(n\) such that there is a \(K_k\)-free graph of order \(n\) for which any three coloring of its vertices (resp. edges) yields a monochromatic copy of \(K_t\). In this note, an algorithm for testing \((t,t,\ldots,t;k)\) in cyclic graphs is presented and it is applied to find new upper bounds for some vertex or edge Folkman numbers. By using this method, we obtain \(F_v(3,3,3;4) \leq 66\), \(F_v(3,3,3;5) \leq 24\), which leads to \(F_v(6,6,6;7) \leq 726\), and \(F_v(3,3,3;8) \leq 727\).

As usual, \(K_{m,n}\) denotes the complete bipartite graph with parts of sizes \(m\) and \(n\). For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j: 0 \leq i \leq n-1, j = i,i+1, \ldots, i+k-1 \pmod{n}\}\). A spider is a tree with at most one vertex of degree more than two, called the \({center}\) of the spider. A leg of a spider is a path from the center to a vertex of degree one. Let \(S_l(t)\) denote a spider of \(l\) legs, each of length \(t\). An \(H\)-decomposition of a graph \(G\) is an edge-disjoint decomposition of \(G\) into copies of \(H\). In this paper, we investigate the problems of \(S_l(2)\)-decompositions of complete bipartite graphs and crowns, and prove that: (1) \(K_{n,tl}\) has an \(S_l(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\), \(n \geq 2l\) if \(t = 1\), and \(n \geq 1\) if \(t \geq 2\), (2) for \(t \geq 2\) and \(n \geq tl\), \(C_{n,tl}\) has an \(S_l(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\), and (3) for \(n \geq 3t\), \(C_{n,tl}\) has an \(S_3(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\) and \(n \equiv 0 \pmod{4}\) if \(t = 1\).

In this paper, we extend the study on packing complete graphs \(K_v\) with \(6\)-cycles. Mainly, we obtain the maximum packing of \(K_v – L\) and a leave, where \(L\) is a vertex-disjoint union of cycles in \(K_v\).

For a vertex \(v\) of a graph \(G\), the unlabeled subgraph \(G-v\) is called a \({card}\) of \(G\). We prove that the connectedness of an \(n\)-vertex graph \(G\) and the presence of isolated vertices in \(G\) can be determined from any collection of \(n-2\) of its cards. It is also proved that if two graphs on \(n \geq 6\) vertices with minimum degree at least two have \(n-2\) cards in common, then the numbers of edges in them differ by at most one.

Let \(G\) be a connected cubic graph embedded on a surface \(\Sigma\) such that every face is bounded by a cycle of length \(6\). By Euler formula, \(\Sigma\) is either the torus or the Klein bottle. The corresponding graphs are called toroidal polyhex graphs and Klein-bottle polyhex graphs, respectively. It was proved that every toroidal polyhex graph is hamiltonian. In this paper, we prove that every Klein-bottle polyhex graph is hamiltonian. Furthermore, lower bounds for the number of Hamilton cycles in Klein-bottle polyhex graphs are obtained.

The matching preclusion number of a graph \(G\), denoted by \(mp(G)\), is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. Besides its theoretical linkage with conditional connectivity and extremal graph theory, the matching preclusion number serves as a measure of robustness in interconnection networks. In this paper, we develop general properties related to matchings in the Cartesian product of graphs, enabling us to establish the matching preclusion number for various interconnection (product) networks, specifically: hyper Petersen, folded Petersen, folded Petersen cube, hyperstar, star-cube, and hypercube. Furthermore, we show that the Cartesian product of graphs operation inherits the matching preclusion number optimality from factor graphs of even order, reinforcing the Cartesian product as a desirable network-synthesizing operator.

This paper proves that the graphic matroids with at least two edges and no isolated vertices coincide with the class of complete \(k\)-partite graphs, where, when \(k \leq 3\), no partition class has size one. It also shows that a simple rank-\(r\) binary matroid \(M\) has every two elements in a \(4\)-circuit if \(|E(M)| \geq 2^{r-1} + 2\).

Multi-sender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we constructed one multi-sender authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of this codes are also computed.

Let \(G\) be a graph with vertex set \(V\). A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V-D\) has a neighbor in \(V-D\). The minimum cardinality of a total restrained dominating set of \(G\) is called the total restrained domination number of \(G\), denoted by \(\gamma_{tr}(G)\). Cyman and Raczek \((2006)\) showed that if \(G\) is a connected graph of order \(n\) and minimum degree \(\delta\) such that \(2 \leq \delta \leq n-2\), then \(\gamma_{tr}(G) \leq n-\delta\). In this paper, we first introduce the concept of max-min total restrained domination number, denoted by \(\gamma_{tr}^M(G)\), of \(G\), and extend the above result by showing that \(\gamma_{tr}^M(G) \leq \gamma_{tr}(G) \leq n-\delta\). We then proceed to establish that \((1)\) \(\gamma_{tr}^M(G) \leq n-2\delta\) if \(n \geq 11\) and \(G\) contains a cut-vertex, and \((2)\) \(\gamma_{tr}(G) \leq n-4\) if \(n \geq 11\) and \(\delta \geq 2\).

In response surface analysis, it is generally assumed that the observations are independent and there is no effect of neighbouring units. But under the situation when the units are placed linearly with no gaps, the experimental units may experience neighbour or overlap effects from neighbouring units. Hence, for proper specification it is important to include the neighbour effects in the model. First order response surface mode! with neighbour effects from immediate left and right neighbouring units has been considered here and the conditions have been derived for the orthogonal estimation of coefficients of this model. The variance of estimated response has also been obtained and conditions for first order response surface model with neighbour effects to be rotatable have been obtained. A method of obtaining designs satisfying the derived conditions has been proposed. A first order rotatable design with neighbour effects using half replicate of \(2^3\) has also been given.

In [J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A \(118(2011) 2056-2058]\), Guo and Wang proposed a new model for disjunct matrices. As a generalization of Guo-Wang’s designs, we obtain a
new family of pooling designs. Our designs and Guo-Wang’s designs have the same numbers of items and pools, but the error-tolerance property of our design is better than that of Guo-Wang’s designs under some conditions.

A \({vertex \;irregular\; total \;labeling}\) \(\sigma\) of a graph \(G\) is a labeling of vertices and edges of \(G\) with labels from the set \(\{1, 2, \ldots, k\}\) in such a way that for any two different vertices \(x\) and \(y\), their weights \(wt(x)\) and \(wt(y)\) are distinct. The \({weight}\) \(wt(x)\) of a vertex \(x\) in \(G\) is the sum of its label and the labels of all edges incident with \(x\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total labeling is called the \({total \;vertex\; irregularity \;strength}\) of \(G\). In this paper, we study the total vertex irregularity strength for two families of graphs, namely Jahangir graphs and circulant graphs.

The Sum-Balaban index is defined as
\[SJ(G) = \frac{|E(G)|}{\mu+1} \sum\limits_{uv \in E(G)} \frac{1}{\sqrt{D_G(u)+D_G(v)}}\],
where \(\mu\) is the cyclomatic number of \(G\) and \(D_G(u)=\sum_{u\in V(G)}d_G(u,v)\). In this paper, we characterize the tree with the maximum Sum-Balaban index among all trees with \(n\) vertices and diameter \(d\). We also provide a new proof of the result that the star \(S_n\) is the graph which has the maximum Sum-Balaban index among all trees with \(n\) vertices. Furthermore, we propose a problem for further research.

A connected graph \(G = (V, E)\) is called a quasi-unicycle graph if there exists \(v_0 \in V\) such that \(G – v_0\) is a unicycle graph. Denote by \(\mathcal{G}(n, d_0)\) the set of quasi-unicycle graphs of order \(n\) with the vertex \(v_0\) of degree \(d_0\) such that \(G – v_0\) is a unicycle graph. In this paper, we determine the maximum spectral radii of quasi-unicycle graphs in \(\mathcal{G}(n, d_0)\).

Let \(Diag(G)\) and \(D(G)\) be the degree-diagonal matrix and distance matrix of \(G\), respectively. Define the multiplier \(Diag(G)D(G)\) as the degree distance matrix of \(G\). The degree distance of \(G\) is defined as \(D'(G) = \sum_{x \in V(G)} d_G(x) D(x)\), where \(d_G(u)\) is the degree of vertex \(x\), \(D_G(x)=\sum_{u\in V(G)}d_G(u,x)\) and \(d_G(u,x)\) is the distance between \(u\) and \(v\). Obviously, \(D'(G)\) is also the sum of elements of the degree distance matrix \(Diag(G)D(G)\) of \(G\). A connected graph \(G\) is a cactus if any two of its cycles have at most one common vertex. Let \(\mathcal{G}(n,r)\) be the set of cacti of order \(n\) and with \(r\) cycles. In this paper, we give the sharp lower bound of the degree distance of cacti among \(\mathcal{G}(n,r)\), and characterize the corresponding extremal cactus.

We introduce the concept of molds, which together with an appropriate weight function, gives all the information of a regular tournament. We use the molds to give a shorter proof of the characterization of domination graphs than the one given in \([4, 5]\), We also use the molds to give a lower and an upper bound of the dichromatic number for all regular tournaments with the same mold.

In this paper, we prove that every countable set of formulas of the propositional logic has at least one equivalent independent subset. We illustrate the situation by considering axioms for Boolean algebras; the proof of independence we give uses model forming.

In this paper, we introduce a new type of graph labeling known as \({super\; mean \;labeling}\). We investigate the super mean labeling for the Complete graph \(K_n\), the Star \(K_{1,n}\), the Cycle \(C_{2n+1}\), and the graph \(G_1 \cup G_2\), where \(G_1\) and \(G_2\) are super mean graphs, as well as some standard graphs.

The \({corona}\) of two graphs \(G\) and \(H\), written as \(G \odot H\), is defined as the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and joining by an edge the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae of the (modified) Schultz and Zagreb indices in the corona of two graphs.

A geodetic (resp. monophonic) dominating set in a connected graph \(G \) is any set of vertices of \(G\) which is both a geodetic (resp.monophonic) set and a dominating set in \(G\). This paper establishes some relationships between geodetic domination and monophonic domination in a graph. It also investigates the geodetic domination and monophonic domination in the join, corona and composition of
connected graphs.

Let \(G\) and \(F\) be graphs. If every edge of \(G\) belongs to a subgraph of \(G\) isomorphic to \(F\), and there exists a bijection \(\lambda: V(G) \bigcup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that the set \(\{\sum_{v\in V(F’)}\lambda(v)+\sum_{e\in E(f’)}\lambda(e):F’\cong F,F’\subseteq G\}\) forms an arithmetic progression starting from \(a\) and having common difference \(d\), then we say that \(G\) is \((a,d)\)-\(F\)-antimagic. If, in addition, \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then \(G\) is \emph{super} \((a,d)\)-\(F\)-antimagic. In this paper, we prove that the grid (i.e., the Cartesian product of two nontrivial paths) is super \((a,1)\)-\(C_4\)-antimagic.

Given a (directed) graph \(G = (V,A)\), the induced subgraph of \(G\) by a subset \(X\) of \(V\) is denoted by \(G[X]\). A graph \(G = (V, A)\) is a \({tournament}\) if for any distinct vertices \(x\) and \(y\) of \(G\), \(G[\{x, y\}]\) possesses a single arc. With each graph \(G = (V,A)\), associate its \({dual}\) \(G^* = (V, A^*)\) defined as follows: for \(x,y \in V\), \((x,y) \in A^*\) if \((y,x) \in A\). Two graphs \(G\) and \(H\) are \({hemimorphic}\) if \(G\) is isomorphic to \(H\) or to \(H^*\). Moreover, let \(k > 0\). Two graphs \(G = (V,A)\) and \(H = (V,B)\) are \({k\;-hemimorphic}\) if for every \(X \subseteq V\), with \(|X| \leq k\), \(G[X]\) and \(H[X]\) are hemimorphic. A graph \(G\) is \({k\;-forced}\) when \(G\) and \(G^*\) are the only graphs \(k\)-hemimorphic to \(G\). Given a graph \(G = (V,A)\), a subset \(X\) of \(V\) is an \({interval}\) of \(G\) provided that for \(a,b \in X\) and \(x \in V\setminus X\), \((a,x) \in A\) if and only if \((b,x) \in A\), and similarly for \((x,a)\) and \((x,b)\). For example, \(\emptyset\), \(\{x\}\), where \(x \in V\), and \(V\) are intervals called trivial. A graph \(G = (V, A)\) is \({indecomposable}\) if all its intervals are trivial. Boussairi, Tle, Lopez, and Thomassé \([2]\) established the following duality result. An indecomposable graph which does not contain the graph \(({0, 1, 2}, {(0, 1), (1,0), (1,2)})\) and its dual as induced subgraphs is \(3\)-forced. A simpler proof of this theorem is provided in the case of tournaments and also in the general case. The \(3\)-forced graphs are then characterized.